With the exception of terminology specific to computers, "function" in computer science refers to mathematical operations. The computability of a function is a crucial quality in this field. Many models of computing have been created to give this idea — and the related concept of algorithm — exact meanings. The traditional ones include general recursive functions, lambda calculus, and Turing machines. The central proposition of computability theory states that all alternative models of computation that have ever been suggested either define the same set of computable functions or a smaller set than what is defined by these three models of computation. The Church-Turing thesis asserts that every definition of a computable function that is philosophically acceptable also defines the same functions.

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The partial functions from integers to integers known as general recursive functions can be defined through the use of the operators, constant functions, successors, projection functions, minimization, composition and primitive recursion.

They are only defined for functions from integers to integers, but because of the following characteristics, they can model any computable function :

Every series of symbols may be represented as a sequence of bits, and a bit sequence can be read as the binary representation of an integer. A computation is the manipulation of finite sequences of symbols ( number digits, formulas, etc. ).

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Functional programming's theoretical foundation is lambda calculus, a theory that specifies computable functions without relying on set theory. It is made up of terms that are either variables, function definitions ( often known as "terms" ), or terms to which functions are applied. The axioms of the theory, the equivalence, the reduction, and the conversion, can be seen as principles of computing and are used to manipulate terms. 

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Domain and codomain of a function are not part of lambda calculus in its original form. In typed lambda calculus, they have essentially been introduced as type in the theory. Compared to untyped lambda calculi, most types of typed lambda calculi can define fewer functions.

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According to this definition, a "graph" is a collection of item pairs. The functions from the real numbers to themselves are the most appropriate to graphs in the sense of diagrams. All functions can be defined by sets of pairs, but creating a graphic for functions between sets may not be feasible ( such as sets of matrices ).

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Functions have varied applications that span across Astronomy, Deep Oceanography, Geospatial Information Systems, Earth Seismology, Physical Sciences, Astronomical Measurements, Machine Learning, Acoustics Technology, navigational sciences, Artificial Intelligence, Electronics Technology, Electrical Technology , Air Sensors & Detection Systems,  Informatics Practices , Radars,  Computers & Communication Technology , Meteorology,  Information & Communication Technology , Under Water Technologies and many other scientific fields.

This is a result of the extensionality axiom, which states that two sets are identical if and only if they include the same members. Some authors exclude codomain from a function's definition, in which case the idea of equality must be treated carefully.

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Although most of the mathematical operations that are encountered in elementary courses are considered elementary in this context, some elementary operations, such as those involving the roots of complex polynomials, are not considered elementary in the common sense.

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Sets :

The mathematical representation of a collection of unique items is called a set ; A set contains elements or members, which can be any type of mathematical object, including variables, other sets, numbers, symbols, points in space, lines, and other geometric shapes. A set with a single element is a singleton, while a set with no elements is an empty set. A set can either be infinite or have a finite number of elements. If all of the elements in two sets are identical, then the sets are equal. 

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Modern mathematics is rife with sets. In fact, since the first part of the 20th century, Zermelo-Fraenkel set theory has been the de facto method for constructing solid foundations for all areas of mathematics. The concept of a set first appeared in mathematics at the end of the nineteenth century. Bernard Bolzano coined the German word for set, Menge, in his book Paradoxes of the Infinite.

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A set is a collection of definite, distinct objects of our perception or thought that are referred to as set elements. The most important property of a set is that it can have elements, also known as members. When two sets have the same elements, they are equal. Sets A and B are equal if every element of A is an element of B and every element of B is an element of A; this property is known as set extensionality.

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The simple concept of a set has proven extremely useful in mathematics, but paradoxes arise when no restrictions on how sets can be constructed are imposed. Russell's paradox demonstrates that the "set of all sets that do not contain themselves" does not exist. Cantor's paradox demonstrates that "the set of all sets" does not exist. A set, according to naive set theory, is any well-defined collection of distinct elements, but problems arise due to the ambiguity of the term well-defined.

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The qualities of sets have been defined by axioms in further attempts to address these dilemmas since the original formulation of naive set theory. The idea of a set is treated as a fundamental concept in axiomatic set theory. The axioms serve as a foundation from which first-order logic can be used to infer the validity or falsity of specific mathematical propositions ( statements ) regarding sets. Yet, it is impossible to utilise first-order logic to demonstrate that any such specific axiomatic set theory is paradox-free, in accordance with Gödel's incompleteness theorems.

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Sets are frequently identified in mathematical publications by capital letters in italics, such as A, B, and C. A set may also be referred to as a collection or family, particularly if its components are also sets. A set is defined by listing each member of it within curly brackets and commas in roster or enumeration notation. The ordering of the elements in roster notation is meaningless because the only thing that matters in a set is whether each element is present or not ( in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters ).

The list of members can be condensed for sets with several items, especially those that adhere to an implicit pattern, by using an ellipsis. A set with an unlimited number of elements is known as an infinite set. An ellipsis, or two ellipses, are used to denote an infinite set in roster notation, indicating that the list goes on forever.

The set - builder notation describes a set as a pick from a bigger set that is based on an elemental condition. The description can be read as "F is the set of all integers n such that n is an integer in the range from 0 to 19 inclusive." The vertical bar "|" in this notation denotes "such that." Some authors choose to omit the vertical bar in favour of a colon, ":". 

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A rule is used in an intensional definition to determine membership. Examples include set-builder notation-based definitions and semantic definitions.
A set's elements are listed in an extensional definition's description of the set. Enumerative definitions are another name for them. A roster with an ellipsis is an example of an ostensive definition, which uses examples of elements to explain a collection.

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The only set that is empty or null contains no members. It is represented as an emptyset. A is referred to as being a subset of B or contained in B, if every element of set A is also in set B. A collection of sets is graphically represented by an Euler diagram, where each set is represented by a planar region with its members inside, enclosed by a loop.

The region representing A is entirely contained within the region representing B if A is a subset of B. When two sets do not share any elements, the regions do not overlap. In comparison, a Venn diagram is a graphical representation of n sets where the n loops split the plane into n zones. Mathematicians commonly refer to certain sets of mathematical significance, and as a result, they have unique names and notational norms to help them be recognised.

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The number of elements in each of the aforementioned sets of integers is limitless. Each of the sets below it is a subset of the others. Superscript plus and minus signs are occasionally used to indicate sets of positive or negative numbers. In more technical terms, a function is a special kind of relation, one that connects each element of A to precisely one element of B. It is a rule that assigns each "input" element of A, a "output" that is an element of B.

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In sets, we deal with many types of functions, namely, injective ( or one - to - one ) if it maps any two different elements of A to different elements of B; surjective ( or onto ) if there is at least one element of A that maps to each element of B; and bijective ( or a one - to - one correspondence ) if the function is both injective and surjective. In this case, there are no unpaired elements because each element of A is paired with a unique element of B. 

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A bijective function is also known as a one-to-one correspondence, a surjective function is known as a surjection, and an injective function is known as an injection. The number of members in a set S is its cardinality, indicated by the symbol | S |. For instance, | B | Equals 3 if B = "blue, white, red." Roster notation does not count repeated members, therefore | blue, white, red, blue, white | also equals three. Formally, two sets are said to have the same cardinality if their relationships are one - to - one. The empty set has zero cardinality.

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Some sets have an infinite or limitless list of elements. For instance, the set of natural numbers in math is infinite. In actuality, all of the unique sets of numbers listed in the previous section are limitless. The cardinality of infinite sets is infinite.

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Several infinite cardinalities have different sizes. The fact that the set of real numbers has more cardinality than the set of natural numbers is perhaps one of the most important conclusions from set theory. The term "countable sets" refers to sets whose cardinality is less than or equal to that of "math N"; these sets can either be finite sets or countably infinite sets ( sets with the same cardinality as "math N") ; some authors use the term "countable" to refer to "countably infinite" sets. 

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There is no set with cardinality precisely between the cardinality of the natural numbers and the cardinality of a straight line, according to Georg Cantor's continuum theory, which was first forth in 1878. Paul Cohen demonstrated in 1963 that the continuum hypothesis is independent of the Zermelo - Fraenkel set theory and the axiom of choice that make up the ZFC axiom system.

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Modern mathematics is rife with sets. In abstract algebra, for instance, structures like groups, fields, and rings are sets that are closed under one or more operations. The creation of relations is one of the primary uses of naive set theory. The Cartesian product A x B is a subset of a relation between domains A and B.

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The set of all S-subsets is known as the power set of a set. As they are both subsets of S, the empty set and S are both parts of the power set of S. A set of nonempty subsets of a set S is referred to as a partition if each member x in S is contained in exactly one of these subsets. Since no two sets of the partition have any elements in common, the subsets are pairwise disjoint, and S is the union of all the subsets of the partition.

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Assume a universal set U has been fixed, which contains all of the elements under discussion, and that A is a subset of U. The set of all items (from U) that do not belong to A is known as the complement of A. It could be written as Ac or A′. Given any two sets A and B, respectively, their union is the collection of all objects that belong to A or B, or both is referred to as A U B. Their intersection is the collection of all objects that belong to both A and B and is known as A intersection B. 

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According to one of De Morgan's laws, (A ∪ B)′ = A′ ∩ B′. (that is, the elements outside the union of A and B are the elements that are outside A and outside B).

The product of the cardinalities of A and B is the cardinality of A x B. When A and B are both finite, this is a basic fact. Multiplication of cardinal numbers is defined to make this true when one or both are infinite.)

Any set's power set is transformed into a Boolean ring by adding its symmetric difference and multiplying its intersection. A method for counting the elements in a union of two finite sets in terms of the dimensions of the two sets and their intersection is the inclusion-exclusion principle.

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Matrices :

A matrix, sometimes known as matrices, is a rectangular array or table of letters, numbers, or other symbols organised in rows and columns that is used to represent a mathematical object or a characteristic of one.

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Matrix representations of linear mappings without additional details enable explicit computations in linear algebra. As a result, a significant portion of linear algebra involves the study of matrices, and the majority of the characteristics and operations of abstract linear algebra may be described in terms of matrices. The composition of linear maps, for instance, is represented by matrix multiplication.

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Not every matrix has a connection to linear algebra. This is especially true for incidence matrices and adjacency matrices in graph theory. Unless otherwise stated, all matrices in this article that are connected to linear algebra represent linear maps or can be interpreted as such.

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In matrix theory, square matrices — those with the same number of rows and columns — play a significant role. One of the most typical examples of a noncommutative ring is formed by square matrices in a particular dimension. For example, a square matrix is invertible if and only if it has a nonzero determinant, and the eigenvalues of a square matrix are the roots of a polynomial determinant. The determinant of a square matrix is a number connected to the matrix, which is crucial for the study of a square matrix.

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Matrix representations of coordinate changes and geometric transformations ( such as rotations ) are frequently utilised in geometry. Many computing issues in numerical analysis are resolved by matrix reduction.

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The entries of a matrix are a rectangular array of numbers ( or other mathematical objects ). Standard operations like addition and multiplication can be performed on matrices. A matrix over a field F is often just a rectangular array of F's constituent elements. Real and complex matrices are those whose entries are real or complex numbers, respectively.

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This frequently entails computation with matrices with enormous dimensions. The majority of mathematical and scientific disciplines make use of matrices, either directly or indirectly through the use of geometry and numerical analysis.

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The entries or elements of the matrix are its numbers, symbols, or phrases. In a matrix, the terms "rows" and "columns" refer to the horizontal and vertical rows of entries, respectively.

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The quantity of rows and columns in a matrix determines its size. As long as they are positive integers, there is no restriction on how many rows and columns a matrix ( in the conventional sense ) can have. An m n matrix, also known as an m-by-n matrix, is a matrix having m rows and n columns. M and n are referred to as the matrix's dimensions. As an illustration, matrix A above is a 3 by 2 matrix.

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Row vectors are matrices with a single row, and column vectors are matrices with a single column.

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A square matrix is one that has the same number of rows and columns. An infinite matrix is one that has an infinite number of rows, columns, or both. Consider a matrix with no rows or columns, often known as an empty matrix, in some situations, such as computer algebra applications.

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A single-row matrix that is occasionally used to represent a vector.  A  single - column matrix that is occasionally used to represent a vector. A matrix with the same number of rows and columns that is occasionally used to describe a linear transformation from one vector space to another, such as reflection, rotation, or shearing.

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Matrix addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix are some of the fundamental operations that can be used to alter matrices.

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Some matrix operations share common characteristics with operations on numbers, such as addition being commutative, which means that the sum of the matrix is independent of the order of the summands. The transpose is compatible with scalar multiplication and addition. If and only if the number of columns in the left matrix equals the number of rows in the right matrix, we can say that two matrices have been multiplied.

In stark contrast to ( rational, real, or complex ) numbers, whose product is independent of the order of the parts, matrix multiplication is not commutative.

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Several less common operations on matrices that can be regarded as forms of multiplication exist in addition to the standard matrix multiplication just mentioned, such as the Hadamard product and the Kronecker product. These appear when attempting to solve matrix equations like the Sylvester equation.

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Row operations come in three different flavours :

Adding a row to another row is known as row addition. Row multiplication, which is the process of multiplying each row's entries by a non-zero constant; Row switching, which involves a matrix's two rows being switched.

Solving linear equations and locating matrix inverses are two examples of the many applications for these procedures.

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A matrix's submatrix can be created by removing any group of rows and / or columns. By calculating the determinant of specific submatrices, one can determine the minors and cofactors of a matrix.

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A square submatrix called a principal submatrix is created by deleting particular rows and columns. From author to author, the term changes. Some writers define a principal submatrix as a submatrix in which the set of remaining row indices and column indices are the same. According to some authors, a principal submatrix is one in which the top k rows and columns, for any value of k, are the only ones left. This kind of submatrix is also referred to as a principal principal submatrix.

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Matrix multiplication relates to map composition according to the one-to-one connection between matrices and linear maps. The greatest number of the matrix's linearly independent column vectors as well as its maximum number of linearly independent row vectors make up the rank of a matrix A. It is, in essence, the size of the linear map image that is represented by A. According to the rank-nullity theorem, the number of columns in a matrix is equal to the dimension of its kernel times its rank.

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A is referred to as an upper triangular matrix if all of its entries below the main diagonal are zero. Similar to this, A is referred to as a lower triangular matrix if all of its elements above the primary diagonal are zero. A is referred to as a diagonal matrix if all entries other than the main diagonal are zero.

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It is both a unique form of diagonal matrix and a square matrix of order n. It is known as an identity matrix because it multiplies a matrix without changing it. A scalar matrix is a nonzero scalar multiple of an identity matrix. The scalar matrices form a group under matrix multiplication that is isomorphic to the multiplicative group of nonzero elements of the field if the matrix entries are taken from a field.

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Hermitian matrices, which satisfy A = A, are frequently used in place of symmetry in complex matrices. The star or asterisk indicates the matrix's conjugate transpose, which is the transposition of the complex conjugate of A.

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Real symmetric matrices and complex Hermitian matrices both have an eigenbasis according to the spectral theorem, meaning that any vector can be expressed as a linear combination of eigenvectors. All eigenvalues are real in both scenarios. This theorem can be extended to situations in infinite dimensions involving matrices with an unlimited number of rows and columns.

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The symmetric matrix is said to be positive-semidefinite if the quadratic form f only produces non-negative values ( positive or zero ) ( or negative -semidefinite if only non - positive values are produced ). The matrix is indefinite precisely when it is neither positive - semidefinite nor negative - semidefinite.

If and only if all of a symmetric matrix's eigenvalues are positive, the matrix is positive-definite, which also means that it is invertible. Two 2 - by - 2 matrix options are shown in the table to the right.

The terminology and outcome are the same for complex matrices, with symmetric matrix, hermitian form, sesquilinear form, and conjugate transpose replacing, respectively, quadratic form, bilinear form, and transpose.

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A square matrix with real entries whose columns and rows are orthogonal unit vectors is known as an orthogonal matrix. A matrix A is equivalently said to be orthogonal if its transpose equals its inverse. 

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An orthogonal matrix with determinant + 1 is a specific orthogonal matrix. Every orthogonal matrix with determinant +1 is a pure rotation without reflection as a linear transformation, which keeps the orientation of the transformed structure intact. Conversely, every orthogonal matrix with determinant -1 reverses the orientation by combining a pure reflection and a ( possibly null ) rotation. The identity matrices are simple rotations with angle zero and have determinant 1.

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A unitary matrix is an orthogonal matrix's complex analogue.

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The total of the diagonal entries makes up a square matrix A's trace, or tr(A). Despite the fact that, as was already mentioned, matrix multiplication is not commutative, the trace of the product of two matrices is unaffected by the factors' order.

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Thus, the trace of the product of more than two matrices is independent of cyclic permutations of the matrices. Nevertheless, this does not generally hold true for arbitrary permutations ( such as, for instance, tr(ABC) ≠ tr(BAC) ). Moreover, a matrix's trace and its transpose are equivalent.

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A integer that encodes specific matrix characteristics is known as the determinant of a square matrix A, represented by det(A) or |A|. If and only if a matrix's determinant is nonzero, the matrix is invertible.

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Three by three matrix determinants have six terms ( rule of Sarrus ). The Leibniz formula, which is longer, applies these two principles to all dimensions. A square matrix's product's determinant is equal to the sum of its determinants. 

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The determinant is unaffected by adding a multiple of any row to another row or a multiple of any column to another column. The determinant is affected by changing two rows or two columns by multiplying it by 1. Any matrix may be converted into a lower ( or upper ) triangular matrix using these operations, and for such matrices, the determinant is equal to the product of the entries on the major diagonal, offering a way to figure out the determinant of any matrix. The determinant is finally expressed using the Laplace expansion in terms of minors, or determinants of smaller matrices.

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The determinant of a 1 - by - 1 matrix, which is its only entry, or even the determinant of a 0 - by - 0 matrix, which is 1, can be used to start a recursive definition of determinants that can be seen to be equivalent to the Leibniz formula. Cramer's rule, which states that the division of the determinants of two related square matrices equals the value of each of the system's variables, can be utilised to use determinants to solve linear equations.

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Many methods can be used to do matrix calculations. Both direct algorithms and iterative methods can be used to tackle a wide variety of issues. By identifying a series of vectors xn that converge to an eigenvector as n approaches towards infinity, for instance, it is possible to determine the eigenvectors of a square matrix. 

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The effectiveness and precision of each algorithm should be evaluated in order to select the one that is best suited for solving each unique challenge. Numerical linear algebra is the field that studies these topics. The complexity of algorithms and their numerical stability are the two key factors, as they are in other numerical circumstances.

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Finding upper limits or estimations of how many basic operations, like additions and scalar multiplications, are required to execute a particular algorithm, such the multiplication of matrices, is how complexity of an algorithm is determined. With the definition provided above, calculating the matrix product of two n-by-n matrices requires n3 multiplications because n multiplications are required for each of the n2 items in the product.

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This "naive" algorithm is outperformed by the Strassen algorithm, which only requires n2.807 multiplications. A more sophisticated method incorporates particular computer device features.

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In many real-world scenarios, more details about the matrices at play are known. Sparse matrices, or matrices with the majority of their entries being zero, are an important instance. For example, the conjugate gradient method is a specially tailored methodology for solving linear systems Ax = b for sparse matrices A.

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A numerically stable algorithm is one in which small variations in the input values do not result in significant variations in the output. The conditioning of linear algebraic issues, like calculating a matrix's inverse, can be captured by a matrix's norm.

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While most programming languages for computers enable arrays, they lack built - in commands for matrices. Alternatively, almost all widely used programming languages offer matrix operations on arrays through external libraries that are readily available. One of the first numerical applications of computers was matrix manipulation. Since its second edition implementation in 1964, the original Dartmouth BASIC contained built-in commands for matrix arithmetic on arrays. Certain engineering desktop computers, such the HP 9830, included ROM cartridges with matrices-specific BASIC instructions as early as the 1970s. Many mathematical programmes can be used to help manipulate matrices, and certain computer languages like APL were created for the purpose.

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Matrix rendering can be done in a number of different ways to make it more user-friendly. Common names for them include matrix decomposition and matrix factorization techniques. The appeal of all these methods is that they maintain certain matrix properties, such as determinant, rank, or inverse, making it possible to calculate these values after the transformation, or that for specific types of matrices, certain matrix operations are algorithmically simpler to perform.

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Lower ( L ) and upper triangular matrices combine to form the LU decomposition factor matrices ( U ). Once this decomposition has been calculated, the straightforward forward and reverse substitution method can be used to solve linear systems more effectively. Similarly, it is simpler to calculate triangular matrices' inverses algorithmically. Similar algorithms include the Gaussian elimination, which converts any matrix to row echelon form. In both cases, the matrix is multiplied by appropriate elementary matrices, which entails permuting the rows or columns or adding multiples of one row to another row. Any matrix A can be expressed using singular value decomposition as the product UDV*, where U and V are unitary matrices and D is a diagonal matrix.

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It is significantly simpler to determine the power of a diagonal matrix by obtaining the corresponding powers of the diagonal elements than it is to exponentiate for A. In order to solve linear differential equations, matrix logarithms, and matrix square roots, it is usually necessary to compute the matrix exponential eA. Other techniques, like the Schur decomposition, can be used to prevent numerically ill-conditioned situations.

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Several generalisations can be made for matrices. While linear algebra codifies the characteristics of matrices in the concept of linear maps, abstract algebra uses matrices with entries in more generic fields or even rings. It is feasible to think about matrices with an endless number of rows and columns. Tensors are another expansion, which can be thought of as higher-dimensional arrays of numbers. Tensors differ from vectors and matrices, which are both rectangular or two-dimensional arrays of numbers, in that vectors are frequently realised as sequences of numbers. Matrix groups are formed by matrices when specific conditions are met. Matrix rings are rings that matrices can generate under specific circumstances. Despite the fact that the product of matrices is not always commutative, some matrices can nonetheless create fields known as matrix fields.

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The matrices with real or complex number elements are the subject of this article. In contrast to real or complex numbers, matrices can be thought of with far more diverse types of entries. Any field, i.e., a set where addition, subtraction, multiplication, and division operations are specified and behave appropriately, may be used in place of R or C as a first step towards generalisation. Examples include rational numbers or finite fields. Matrix over finite fields are used, for instance, in coding theory. Wherever eigenvalues are taken into account, since they are polynomial roots, they may only exist in a field that is greater than the matrix's elements; for example, they may be complex in the case of a matrix with real entries.

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It is then possible to consider each square matrix to have a complete set of eigenvalues by reinterpreting the entries of a matrix as components of a bigger field (for instance, by viewing a real matrix as a complex matrix whose entries happen to be all real). As an alternative, only matrices having entries in an algebraically closed field, like C, can be taken into account right away.

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In mathematics more generally, matrices containing entries in a ring R are frequently utilised. In that a division operation is not required, rings are a more broad concept than fields. In this situation as well, matrices can be added to and multiplied in the same ways. All square n-by-n matrices over R are represented by the set M(n, R) (sometimes written Mn(R) ), which is a ring known as the matrix ring and is isomorphic to the endomorphism ring of the left R-module Rn. The ring M(n, R) is also an associative algebra over R if the ring R is commutative, or if its multiplication is commutative. The Leibniz formula can still be used to define the determinant of square matrices over a commutative ring R.

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Such a matrix generalises the case over a field F, where every nonzero element is invertible, and is invertible if and only if its determinant is invertible in R. [60] Supermatrices are matrices over superrings.

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Not all entries in matrices are placed in the same ring, or even in any ring at all. Block matrices, which can be thought of as matrices whose entries are matrices themselves, are one unique but typical example. The entries do not have to be square matrices or rings, but their widths must meet certain compatibility requirements.

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The category of all matrices with entries in a field k and multiplication as composition is identical to the category of finite-dimensional vector spaces and linear mappings over this field, which restates these facts more intuitively.

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More generally, the R-linear maps between the free modules Rm and Rn for any ring R with unity can be represented as the set of m x n matrices. These mappings can be composed when n = m, which results in the matrix ring of n x n matrices, which represents the endomorphism ring of Rn.

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A group is a mathematical structure made up of a collection of items and a binary operation, which, under specific conditions, combines any two objects into a third. A matrix group is one in which the objects are matrices and the group operation is multiplication. The most general matrix groups—known as general linear groups—are those composed of all invertible matrices of a given size because a group's elements must all be invertible.

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More matrix groups can be defined using any attribute of matrices that is maintained by matrix products and inverses. For instance, a subgroup of matrices with a certain dimension and a determinant of 1 is (that is, a smaller group contained in) is called a special linear group.

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Determinant 1 or 1 exists in every orthogonal matrix. Special orthogonal group is the name of the subgroup formed by orthogonal matrices with determinant 1. Consider the regular representation of the symmetric group to see that any finite group is isomorphic to a matrix group. Matrix groups, which are rather well understood, can be used to study general groups using representation theory.

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It is also possible to take into account matrices that have an endless number of rows and/or columns despite the fact that because they are infinite objects, they cannot be explicitly written down. All that requires is that there is a clearly defined entry for each element in the set indexing rows and each element in the set indexing columns ( these index sets need not even be subsets of the natural numbers). While it is still possible to describe the fundamental operations of addition, subtraction, scalar multiplication, and transposition without any issues, matrix multiplication may require endless summations to define the resulting entries, and these are not often defined.

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For the following reason, only matrices with all of their columns having a limited number of nonzero entries can be utilised to represent linear maps when using infinite matrices. It is necessary to choose bases for both spaces in order for a matrix A to describe a linear map f: VW. By definition, this means that every vector in the space can be written as a (finite) linear combination of basis vectors, and that only a finite number of entries vi are nonzero when written as a (column) vector v of coefficients.

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Now, the columns of A describe the images by f of distinct basis vectors of V in the basis of W. But, this description is only relevant if the number of nonzero entries in these columns is finite. Yet, there is no restriction on the rows of A because every one of its entries, even if it is given as an infinite sum of products, contains only a finite number of nonzero terms and is thus properly defined. In the product Av, there are only a finite number of nonzero coefficients of v involved. Also, this amounts to creating a linear combination of A's columns that only involves a finite number of them in reality.

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Products of two matrices of the specified type are well defined, of the same type, and correspond to the composition of linear maps (assuming that the column-index and row-index sets match).

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The requirement of row or column finiteness can be removed if R is a normed ring. Finite sums can be replaced by totally convergent series once the norm has been established. For instance, a ring is formed by the matrices whose column sums are absolutely convergent sequences. Similar to this, a ring is formed by matrices whose row sums are absolutely convergent series.

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Moreover, infinite matrices can be used to define operators on Hilbert spaces, where issues with convergence and continuity recur, necessitating the imposition of further restrictions. The more powerful and abstract tools of functional analysis can be utilised in place of matrices, whose explicit point of view tends to obscure the issue.

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A matrix with zero rows, columns, or both is referred to as an empty matrix. When dealing with maps involving the zero vector space, empty matrices are helpful. A 3-by-3 zero matrix, for instance, corresponds to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix if A is a 3-by-0 matrix and B is a 0-by-3 matrix. Although empty matrices can be created and used for computation, there is no standard notation for them. Regarding the empty product appearing in the Leibniz formula for the determinant as 1, the determinant of the 0-by-0 matrix is 1.

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Matrix-based structures have many uses in both mathematics and other areas. Some of them only benefit from a set of integers' compact matrix representation. For instance, in game theory and economics, the reward matrix encodes the payout for two players based on which option they select from a predetermined (limited) set of options. Document-term matrices like the tf-idf are used in text mining and automatic thesaurus creation to keep track of the frequency of particular terms across several documents.

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Complex number addition and multiplication, as well as matrix multiplication, are equivalent. For instance, 2-by-2 rotation matrices, as mentioned before, represent multiplication with a complex number having absolute value 1. Quaternions and Clifford algebras in general are susceptible to a similar interpretation.

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Matrix-based encryption methods from the past, such the Hill cypher, were very popular. These codes are, nevertheless, rather simple to crack because of the linear nature of matrices. A three-dimensional object can be projected onto a two-dimensional screen to simulate a theoretical camera observation. Computer graphics uses matrices to describe objects and to calculate transformations of objects using affine rotation matrices. The study of control theory requires a thorough understanding of matrices over polynomial rings.

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Since the application of quantum theory to analyse molecular bonding and spectroscopy, chemistry has used matrices in a variety of ways. Examples include the Fock matrix and the overlap matrix, which are utilised in the Hartree-Fock method to solve the Roothaan equations and produce the molecular orbitals.

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Matrix-based structures have many uses in both mathematics and other areas. Some of them only benefit from a set of integers' compact matrix representation. For instance, in game theory and economics, the reward matrix encodes the payout for two players based on which option they select from a predetermined (limited) set of options. Document-term matrices like the tf-idf are used in text mining and automatic thesaurus creation to keep track of the frequency of particular terms across several documents.

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A fundamental idea in graph theory is the adjacency matrix of a finite graph. It keeps track of which edges connect the graph's vertices. Logical matrices are matrices with only two distinct values (1 and 0 representing, for instance, "yes" and "no," respectively). Information concerning edge distances is contained in the distance (or cost) matrix. Similar ideas can be extended to websites connected by links, cities connected by roads, etc., in which case the matrices tend to be sparse, that is, to have few nonzero entries, unless the connection network is exceptionally dense. Hence, network theory can make use of specially designed matrix methods.

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The second derivatives of a differentiable function : R n R with respect to the various coordinate directions make up the Hessian matrix for that function. It encodes data pertaining to the function's local growth behaviour.

If the Hessian matrix is positive definite, then the function has a local minimum. Finding the global minima or maxima of quadratic functions that are closely connected to the ones attached to matrices can be done using quadratic programming .

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The Jacobi matrix of a differentiable map f is another matrix that is widely utilised in geometrical circumstances:

According to the implicit function theorem, f is locally invertible at that point if n > m and the rank of the Jacobi matrix reaches its maximum value, m.

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The matrix of coefficients for the highest-order differential operators in the equation can be used to categorise partial differential equations. This matrix is positive definite for elliptic partial differential equations.

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It significantly affects the range of potential solutions to the enumerated equation. 

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The finite element method is a crucial computational approach for resolving partial differential equations and is frequently used to model intricate physical systems. It makes an effort to approximate a particular equation's solution using piecewise linear functions, where the pieces are selected in accordance with a small enough grid, which may then be transformed into a matrix equation.

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Stochastic matrices are square matrices with entries that are non-negative and add to one and whose rows are probability vectors. To define Markov chains with a finite number of states, stochastic matrices are utilised. The probability distribution for the next position of a particle that is currently in the state that corresponds to the row is given by a row of the stochastic matrix. The eigenvectors of the transition matrices can be used to read the properties of the Markov chain-like absorbing states, or the states that every particle finally reaches.

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Matrix calculations are frequently used in statistics in a variety of ways. The goal of descriptive statistics is to describe data sets, which are frequently represented as data matrices and can then be reduced in size using dimensionality. The mutual variance of numerous random variables is encoded in the covariance matrix.

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It relates to the singular value decomposition of matrices and can be expressed in terms of matrices. Matrix entries in random matrices are chosen at random from appropriate probability distributions, such as the matrix normal distribution. They are used outside of probability theory in fields like number theory and physics.

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Modern physics heavily relies on linear transformations and the symmetries that go along with them. For instance, the behaviour of elementary particles under the spin group is used to categorise them as representations of the Lorentz group of special relativity in quantum field theory. The physical description of fermions, which act as spinors, includes concrete representations involving the Pauli matrices and more generic gamma matrices. There is a group -theoretical representation for the three lightest quarks involving the special unitary group SU; for their calculations, physicists use a practical matrix representation known as the Gell-Mann matrices, which is also used for the SU gauge group that serves as the foundation of the contemporary description of strong nuclear interactions, quantum chromodynamics. In turn, the Cabibbo-Kobayashi-Maskawa matrix.

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The Cabibbo-Kobayashi-Maskawa matrix, on the other hand, expresses the fact that the fundamental quark states that are crucial for weak interactions are different from but linearly connected to the fundamental quark states that describe particles with particular and distinct masses.

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The operators of the theory were represented as infinite - dimensional matrices acting on quantum states in Heisenberg's 1925 original quantum mechanics model. This also goes by the name of matrix mechanics. The density matrix, which describes the "mixed" state of a quantum system as a linear mixture of elementary, "pure" eigenstates, is one example in particular.

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For explaining the scattering experiments that are the foundation of experimental particle physics, another matrix is a crucial tool: The scalar product of outgoing particle states and a linear combination of ingoing particle states can be used to describe collision reactions, such as those that take place in particle accelerators when non - interacting particles approach one another and collide in a small interaction zone to create a new set of non-interacting particles.

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The S-matrix, a matrix that stores all information about potential interactions between particles, provides the linear combination. The description of linearly connected harmonic systems is a general use of matrices in physics. The equations of motion for such systems can be expressed as matrices, with the kinetic term being represented by the mass matrix multiplied by the generalised velocity and the interactions being represented by the force matrix multiplied by the displacement vector. The optimum method for finding solutions is to diagonalize the matrix equation and find the system's eigenvectors, or normal modes. These kinds of methods are essential for understanding the internal vibrations of systems made up of mutually bound component atoms, which make up the internal dynamics of molecules. They are also required to describe oscillations in electrical circuits and mechanical vibrations. They are also required to describe oscillations in electrical circuits and mechanical vibrations.

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Further matrix applications are provided by geometrical optics. The wave aspect of light is not taken into consideration in this approximate theory. As a result, a model is created in which light rays are actually geometrical rays. The action of a lens or reflective element on a given light ray can be described as the multiplication of a two-by-two matrix called a ray transfer matrix analysis when the deflection of light rays by optical elements is small. The matrix encodes the characteristics of the optical element, and the vector's components are the slope and distance from the optical axis of the light ray. There are actually two different types of matrices: a translation matrix that describes how the plane is translated, and a refraction matrix that describes how light is refracted at a lens surface.

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The matrix that results from the product of the matrices of the components accurately describes the optical system, which consists of a combination of lenses and / or reflective elements. 

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In electronics, conventional mesh analysis and nodal analysis result in a set of linear equations that can be expressed as a matrix. Matrix-based descriptions of the behaviour of numerous electrical components are possible. Let A be a two-dimensional vector with elements representing the input voltage v1 and the input current i1 of the component, and let B be a two-dimensional vector with elements representing the output voltage v2 and the output current i2 of the component.

Matrices have been used to solve linear equations for a very long time, but until the 1800s, they were known as arrays. The Nine Chapters on the Mathematical Art, a Chinese work from the 10th to 2nd century BCE that includes the idea of determinants, is the first instance of the use of array methods to solve simultaneous equations. Gerolamo Cardano, an Italian mathematician, published Ars Magna in 1545, introducing the approach to Europe. The same array techniques were employed in 1683 by the Japanese mathematician Seki to resolve simultaneous equations. In his work Elements of Curves from 1659, Dutch mathematician Jan de Witt illustrated transformations using arrays (1659). Gottfried Wilhelm Leibniz conducted over 50 experiments and advertised the use of arrays for storing data or solutions between 1700 and 1710.

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The word "matrix" ( Latin for "womb," "dam" ( non - human female animal kept for breeding ), "source," "origin," "list," and "register," derived from mater — mother ) was first used by James Joseph Sylvester in 1850. Sylvester understood a matrix as an object giving rise to several determinants now known as minors, that is, determinants of smaller matrices that derive from the original one.

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In a dissertation on geometric transformations, Arthur Cayley used matrices rather than the previously used rotated versions of the coefficients under consideration. Instead, he demonstrated that the associative and distributive laws held true by defining operations like addition, subtraction, multiplication and division as transformations of those matrices. Cayley looked into and showed both the commutative and non - commutative natures of matrix addition and multiplication. The use of arrays in early matrix theory was almost entirely restricted to determinants, hence Arthur Cayley's abstract matrix operations were groundbreaking. He played a key role in the proposal of an equation system - free matrix paradigm. Cayley conceived and proved the Cayley-Hamilton theorem in his 1858 book A memoir on the theory of matrices.

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The contemporary bracket notation for matrices was invented by the English mathematician Cuthbert Edmund Cullis in 1913. These ideas were taken further by Eisenstein, who also made the observation that matrix products are non-commutative in today's terms. Cauchy was the first to establish broad generalisations regarding determinants.

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There are many influences on how determinants are studied today. Gauss connected coefficients of quadratic forms and linear maps in three dimensions to matrices as a solution to number-theoretic issues. These ideas were taken further by Eisenstein, who also made the observation that matrix products are non-commutative in today's terms. Cauchy was the first to establish broad generalisations regarding determinants.

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Real numbers make up symmetric matrices' eigenvalues. Jacobi studied "functional determinants," later referred to as Jacobi determinants by Sylvester, which can be used to explain geometric transformations at the local ( or infinitesimal ) level. Kronecker's Vorlesungen über die Theorie der Determinanten and Weierstrass' Zur Determinantentheorie, both published in 1903, were the first works to treat determinants axiomatically, as opposed to earlier more established approaches.  Determinants had by this time been thoroughly established.

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In the aforementioned memoir, Cayley proved the Cayley - Hamilton theorem for 2 x 2 matrices, while Hamilton proved it for 4 x 4 matrices. Several theorems were first proven for smaller matrices alone. Frobenius expanded the theorem to encompass all dimensions when working with bilinear forms ( 1898 ). The Gauss - Jordan elimination was also developed at the end of the 19th century.

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Wilhelm Jordan invented the Gauss-Jordan elimination at the end of the 19th century, generalising a specific example that is today known as the Gauss elimination. Due in part to their employment in classifying the extremely complex number systems of the previous century, matrices came to play a crucial role in linear algebra around the beginning of the 20th century.

The study of matrices with infinitely many rows and columns resulted from the development of matrix mechanics by Heisenberg, Born, and Jordan.

Subsequently, by further expanding functional analytical concepts like linear operators on Hilbert spaces, which, generally speaking, correspond to Euclidean space but with an infinite number of independent directions, von Neumann completed the mathematical formulation of quantum mechanics.

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The term "matrix" is used by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) in reference to their axiom of reducibility. This axiom was put forth as a way to sequentially reduce any function to a lower type.

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The term for any function, regardless of how many variables it has, that has no discernible variables. Then, by using generalisation, any possible function other than a matrix derives from a matrix by taking into account the premise that the function in question is true for all feasible values or for some value of one of the arguments, with the other argument( s ) remaining indeterminate.

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In his 1946 Introduction to Logic, Alfred Tarski coined the word "matrix" to refer to the idea of truth.