What is a matrix in algebra?
Matrix (mathematics)
The size of a matrix is defined by the number of rows and columns that it contains. A matrix with "m" rows and "n" columns is called an "m" × "n" matrix or "m"-by-"n" matrix, while "m" and "n" are called its "dimensions". For example, the matrix A above is a 3 × 2 matrix.
Invertible matrix
In linear algebra, an "n"-by-"n" square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an "n"-by-"n" square matrix B such that
Matrix (mathematics)
A nonzero scalar multiple of an identity matrix is called a "scalar" matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.
Involutory matrix
In mathematics, an involutory matrix is a matrix that is its own inverse. That is, multiplication by matrix A is an involution if and only if A = I. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any nonsingular matrix multiplied by its inverse is the identity.
Moment matrix
In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.)
Diagonal matrix
For an abstract vector space "V" (rather than the concrete vector space formula_8), or more generally a module "M" over a ring "R", with the endomorphism algebra End("M") (algebra of linear operators on "M") replacing the algebra of matrices, the analog of scalar matrices are scalar transformations. Formally, scalar multiplication is a linear map, inducing a map formula_9 (send a scalar "λ" to the corresponding scalar transformation, multiplication by "λ") exhibiting End("M") as a "R"-algebra. For vector spaces, or more generally free modules formula_10, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the center of the endomorphism algebra, and similarly invertible transforms are the center of the general linear group GL("V"), where they are denoted by Z("V"), follow the usual notation for the center.
Matrix (mathematics)
A square matrix A is called "invertible" or "non-singular" if there exists a matrix B such that
where I is the "n"×"n" identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the "inverse matrix" of A, denoted A.
Linear algebra
The "determinant" of a square matrix is a polynomial function of the entries of the matrix, such that the matrix is invertible if and only if the determinant is not zero. This results from the fact that the determinant of a product of matrices is the product of the determinants, and thus that a matrix is invertible if and only if its determinant is invertible.
Matrix (mathematics)
Matrices and matrix multiplication reveal their essential features when related to "linear transformations", also known as "linear maps". A real "m"-by-"n" matrix A gives rise to a linear transformation R → R mapping each vector x in R to the (matrix) product Ax, which is a vector in R. Conversely, each linear transformation "f": R → R arises from a unique "m"-by-"n" matrix A: explicitly, the of A is the "i" coordinate of "f"(e), where e = (0...,0,1,0...,0) is the unit vector with 1 in the "j" position and 0 elsewhere. The matrix A is said to represent the linear map "f", and A is called the "transformation matrix" of "f".
Matrix (mathematics)
A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as . For example, the rotation of vectors in three-dimensional space is a linear transformation, which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Matrix (mathematics)
Sometimes, the entries of a matrix can be defined by a formula such as "a" = "f"("i", "j"). For example, each of the entries of the following matrix A is determined by "a" = "i" − "j". 
In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as A = ["i"-"j"], or A = (("i"-"j")). If matrix size is "m" × "n", the above-mentioned formula "f"("i", "j") is valid for any "i" = 1, ..., "m" and any "j" = 1, ..., "n". This can be either specified separately, or using "m" × "n" as a subscript. For instance, the matrix A above is 3 × 4 and can be defined as A = ["i" − "j"] ("i" = 1, 2, 3; "j" = 1, ..., 4), or A = ["i" − "j"].
Triangular matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called ' if all the entries "above" the main diagonal are zero. Similarly, a square matrix is called ' if all the entries "below" the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is called a diagonal matrix.
Exchange matrix
In mathematics, especially linear algebra, the exchange matrix (also called the reversal matrix, backward identity, or standard involutory permutation) is a special case of a permutation matrix, where the 1 elements reside on the counterdiagonal and all other elements are zero. In other words, it is a 'row-reversed' or 'column-reversed' version of the identity matrix.
Representation theory
There are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group "G" or (associative or Lie) algebra "A" on a vector space "V" is a map
with two properties. First, for any "g" in "G" (or "a" in "A"), the map
is linear (over F). Second, if we introduce the notation "g" · "v" for formula_3 ("g", "v"), then for any "g", "g" in "G" and "v" in "V":
where "e" is the identity element of "G" and "g""g" is the product in "G". The requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (1) is ignored. Equation (2) is an abstract expression of the associativity of matrix multiplication. This doesn't hold for the matrix commutator and also there is no identity element for the commutator. Hence for Lie algebras, the only requirement is that for any "x", "x" in "A" and "v" in "V":
where ["x", "x"] is the Lie bracket, which generalizes the matrix commutator "MN" − "NM".
Essential matrix
or
Non-negative matrix factorization
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix is factorized into (usually) two matrices and , with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.
Logarithm of a matrix
one gets
Zero matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. Some examples of zero matrices are
Unimodular matrix
In mathematics, a unimodular matrix "M" is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix "N" which is its inverse (these are equivalent under Cramer's rule). Thus every equation "Mx" = "b", where "M" and "b" are both integer, and "M" is unimodular, has an integer solution. The unimodular matrices of order "n" form a group, which is denoted formula_1.
Vector space
"Matrices" are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any "m"-by-"n" matrix "A" gives rise to a linear map from "F" to "F", by the following
or, using the matrix multiplication of the matrix with the coordinate vector :
Moreover, after choosing bases of and , "any" linear map is uniquely represented by a matrix via this assignment.
The determinant of a square matrix is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero. The linear transformation of corresponding to a real "n"-by-"n" matrix is orientation preserving if and only if its determinant is positive.
Block matrix
where A is a square matrix; in other words, matrix A is the direct sum of A, …, A. It can also be indicated as A formula_38 A formula_39 A  or  diag(A, A,formula_40, A)  (the latter being the same formalism used for a diagonal matrix).
Any square matrix can trivially be considered a block diagonal matrix with only one block.
Manin matrix
Somewhat surprising fact is that this construction applied to the polynomial algebra "C"["x", ..., "x"] will give not the usual algebra of matrices "Mat" (more precisely algebra of function on it), but much bigger non-commutative algebra of Manin matrices (more precisely algebra generated by elements "M".
More precisely the following simple propositions hold true.
Matrix (mathematics)
The entry in the "i"-th row and "j"-th column of a matrix A is sometimes referred to as the "i","j", ("i","j"), or ("i","j") entry of the matrix, and most commonly denoted as "a", or "a". Alternative notations for that entry are "A"["i,j"] or "A". For example, the (1,3) entry of the following matrix A is 5 (also denoted "a", "a", "A"["1,3"] or "A"):
Algebraic K-theory
is the direct limit of the GL("n"), which embeds in GL("n" + 1) as the upper left block matrix, and formula_28 is its commutator subgroup. Define an "elementary matrix" to be one which is the sum of an identity matrix and a single off-diagonal element (this is different from the definition used in linear algebra). Then Whitehead's lemma states that the group "E"("A") generated by elementary matrices equals the commutator subgroup [GL("A"), GL("A")]. Indeed, the group GL("A")/E("A") was first defined and studied by Whitehead, and is called the Whitehead group of the ring "A".
Augmented matrix
In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
Compound matrix
In linear algebra, a branch of mathematics, a compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to exterior algebras.
Nilpotent matrix
In linear algebra, a nilpotent matrix is a square matrix "N" such that
for some positive integer formula_2. The smallest such formula_2 is sometimes called the index of formula_4.
Idempotent matrix
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix formula_1 is idempotent if and only if formula_2. For this product formula_3 to be defined, formula_1 must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.
Matrix multiplication
Computing matrix products is a central operation in all computational applications of linear algebra. Its computational complexity is (for matrices) for the basic algorithm (this complexity is for the asymptotically fastest known algorithm). This nonlinear complexity means that matrix product is often the critical part of many algorithms. This is enforced by the fact that many operations on matrices, such as matrix inversion, determinant, solving systems of linear equations, have the same complexity. Therefore various algorithms have been devised for computing products of large matrices, taking into account the architecture of computers (see BLAS, for example).
Matrix (mathematics)
Matrices with a single row are called "row vectors", and those with a single column are called "column vectors". A matrix with the same number of rows and columns is called a "square matrix". A matrix with an infinite number of rows or columns (or both) is called an "infinite matrix". In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an "empty matrix".
Linear algebra
Two matrices that encode the same linear transformation in different bases are called similar. Equivalently, two matrices are similar if one can transform one in the other by elementary row and column operations. For a matrix representing a linear map from to , the row operations correspond to change of bases in and the column operations correspond to change of bases in . Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector space, this means that, for any linear map from to , there are bases such that a part of the basis of is mapped bijectively on a part of the basis of , and that the remaining basis elements of , if any, are mapped to zero (this is a way of expressing the fundamental theorem of linear algebra). Gaussian elimination is the basic algorithm for finding these elementary operations, and proving this theorem.
String theory
In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.
Linear algebra
If a basis exists that consists only of eigenvectors, the matrix of on this basis has a very simple structure: it is a diagonal matrix such that the entries on the main diagonal are eigenvalues, and the other entries are zero. In this case, the endomorphism and the matrix are said diagonalizable. More generally, an endomorphism and a matrix are also said diagonalizable, if they become diagonalizable after extending the field of scalars. In this extended sense, if the characteristic polynomial is square-free, then the matrix is diagonalizable.
Manin matrix
Manin matrices is a general and natural class of matrices with not-necessarily commutative elements which admit natural definition of the determinant and generalizations of the linear algebra theorems.
Generalized permutation matrix
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is
Matrix ring
In abstract algebra, a matrix ring is any collection of matrices over some ring "R" that form a ring under matrix addition and matrix multiplication . The set of matrices with entries from "R" is a matrix ring denoted M("R"), as well as some subsets of infinite matrices which form infinite matrix rings. Any subring of a matrix ring is a matrix ring.
Symmetry in mathematics
In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
Supermatrix
In mathematics and theoretical physics, a supermatrix is a Z-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinary field (thought of as a purely even commutative superalgebra).
Matrix (mathematics)
A group is a mathematical structure consisting of a set of objects together with a binary operation, that is, an operation combining any two objects to a third, subject to certain requirements. A group in which the objects are matrices and the group operation is matrix multiplication is called a "matrix group". Since in a group every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.
Matrix (mathematics)
Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any ring; but their sizes must fulfil certain compatibility conditions.
Vectorization (mathematics)
In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of an matrix "A", denoted vec("A"), is the column vector obtained by stacking the columns of the matrix "A" on top of one another:
Matrix field
In abstract algebra, a matrix field is a field with matrices as elements. In field theory we come across two type of fields: finite field and infinite field. There are several examples of matrix fields of finite and infinite order. In general, corresponding to each field of numbers there is a matrix field.
Matrix representation
Matrix representation is a method used by a computer language to store matrices of more than one dimension in memory.
Fortran and C use different schemes for their native arrays. Fortran uses "Column Major", in which all the elements for a given column are stored contiguously in memory. C uses "Row Major", which stores all the elements for a given row contiguously in memory.
LAPACK defines various matrix representations in memory. There is also Sparse matrix representation and Morton-order matrix representation.
According to the documentation, in LAPACK the unitary matrix representation is optimized. Some languages such as Java store matrices using Iliffe vectors. These are particularly useful for storing irregular matrices. Matrices are of primary importance in linear algebra.
Matrix (mathematics)
Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an "m"-by-"n" matrix, x designates a column vector (that is, "n"×1-matrix) of "n" variables "x", "x", ..., "x", and b is an "m"×1-column vector, then the matrix equation
Diagonalizable matrix
In linear algebra, a square matrix formula_1 is called diagonalizable or nondefective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix formula_2 such that formula_3 is a diagonal matrix. If formula_4 is a finite-dimensional vector space, then a linear map formula_5 is called diagonalizable if there exists an ordered basis of formula_4 with respect to which formula_7 is represented by a diagonal matrix. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. A square matrix that is not diagonalizable is called "defective."
Adjacency matrix
For a simple graph with vertex set "V", the adjacency matrix is a square  ×  matrix "A" such that its element "A" is one when there is an edge from vertex "i" to vertex "j", and zero when there is no edge. The diagonal elements of the matrix are all zero, since edges from a vertex to itself (loops) are not allowed in simple graphs. It is also sometimes useful in algebraic graph theory to replace the nonzero elements with algebraic variables.
Zero element
In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being zero. Some examples of zero matrices are
Matrix grammar
where
Identity matrix
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size "n" is the "n" × "n" square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by "I", or simply by "I" if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to "I".) Less frequently, some mathematics books use "U" or "E" to represent the identity matrix, meaning "unit matrix" and the German word "Einheitsmatrix", respectively.
Matrix pencil
In linear algebra, if formula_1 are formula_2 complex matrices for some nonnegative integer formula_3, and formula_4 (the zero matrix), then the matrix pencil of degree formula_3 is the matrix-valued function defined on the complex numbers formula_6
Linear algebra
Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra.
Signature matrix
In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are plus or minus 1, that is, any matrix of the form:
Weyl–Brauer matrices
Let "A" denote the algebra generated by these matrices. By counting dimensions, "A" is a complete 2×2 matrix algebra over the complex numbers. As a matrix algebra, therefore, it acts on 2-dimensional column vectors (with complex entries). These column vectors are the spinors.
Integer matrix
The inverse of an integer matrix formula_3 is again an integer matrix if and only if the determinant of formula_3 is exactly formula_5 or formula_6. Integer matrices of determinant formula_5 form the group formula_8, which has far-reaching applications in arithmetic and geometry. For formula_9, it is closely related to the modular group.
Matrix similarity
Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator:Because of this, for a given matrix "A", one is interested in finding a simple "normal form" "B" which is similar to "A"—the study of "A" then reduces to the study of the simpler matrix "B". For example, "A" is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every matrix is similar to a matrix in Jordan form. Neither of these forms is unique (diagonal entries or Jordan blocks may be permuted) so they are not really normal forms; moreover their determination depends on being able to factor the minimal or characteristic polynomial of "A" (equivalently to find its eigenvalues). The rational canonical form does not have these drawbacks: it exists over any field, is truly unique, and it can be computed using only arithmetic operations in the field; "A" and "B" are similar if and only if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of "A"; these can be immediately read off from a matrix in Jordan form, but they can also be determined directly for any matrix by computing the Smith normal form, over the ring of polynomials, of the matrix (with polynomial entries) (the same one whose determinant defines the characteristic polynomial). Note that this Smith normal form is not a normal form of "A" itself; moreover it is not similar to either, but obtained from the latter by left and right multiplications by different invertible matrices (with polynomial entries).
Matrix analysis
Any matrix can be expressed as a linear combination of basis matrices, which play the role of the basis vectors for the matrix space. For example, for the set of 2×2 matrices over the field of real numbers, "M"(ℝ), one legitimate basis set of matrices is:
Matrix exponential
Note that the exponential of a matrix is always an invertible matrix. The inverse matrix of is given by . This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map
from the space of all "n"×"n" matrices to the general linear group of degree , i.e. the group of all "n"×"n" invertible matrices. In fact, this map is surjective which means that every invertible matrix can be written as the exponential of some other matrix (for this, it is essential to consider the field C of complex numbers and not R).
Alternant matrix
In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternant matrix.
Matrix polynomial
In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial
this polynomial evaluated at a matrix "A" is
where "I" is the identity matrix.
Linear algebra
A symmetric matrix is always diagonalizable. There are non-diagonizable matrices, the simplest being
(it cannot be diagonalizable since its square is the zero matrix, and the square of a nonzero diagonal matrix is never zero).
Rank (linear algebra)
In linear algebra, the rank of a matrix formula_1 is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of formula_1. This, in turn, is identical to the dimension of the space spanned by its rows. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by formula_1. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.
Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as A (also written A′, A, A or A). It is achieved by any one of the following equivalent actions:
Convergent matrix
A matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations () above, with A non-singular, the matrix A can be split, that is, written as a difference
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and :
or in matrix form:
Logarithm of a matrix
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
Matrix splitting
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960.
2 × 2 real matrices
Every real matrix can be interpreted as one of three types of (generalized) complex numbers: standard complex numbers, dual numbers, and split-complex numbers. Above, the algebra of matrices is profiled as a union of complex planes, all sharing the same real axis. These planes are presented as commutative subrings "P". We can determine to which complex plane a given matrix belongs as follows and classify which kind of complex number that plane represents.
Matrix coefficient
In mathematics, a matrix coefficient (or matrix element) is a function on a 
group of a special form, which depends on a linear representation of the group and additional data. For the case of a finite group, matrix coefficients express the action of the elements of the group in the specified representation via the entries of the corresponding matrices.
Distance matrix
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the "distance" being used to define this matrix may or may not be a metric. If there are elements, this matrix will have size . In graph-theoretic applications the elements are more often referred to as points, nodes or vertices.
Algebra over a field
An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order "n" forms a unital algebra since the identity matrix of order "n" is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a (unital) ring that is also a vector space.
Stone–von Neumann theorem
More formally, the group algebra of the Heisenberg group over its field of scalars "K", written , has center , so rather than simply thinking of the group algebra as an algebra over the field , one may think of it as an algebra over the commutative algebra . As the center of a matrix algebra or operator algebra is the scalar matrices, a -structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of -algebras , which is the formal way of saying that it sends the center to a chosen scale.
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The term usually refers to square matrices. An example of a 2-by-2 diagonal matrix is formula_1; the following matrix is a 3-by-3 diagonal matrix:formula_2. An identity matrix of any size, or any multiple of it, will be a diagonal matrix.
Rank (linear algebra)
The matrix
Diagonal matrix
The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size. All other diagonal matrices which are not scalar only commute with other diagonal matrices and not with any matrix unlike scalar matrices. Intuitively, this stems from the fact that scalar matrices are Identity matrices multiplied with scalars.
Matrix function
In mathematics, a matrix function is a function which maps a matrix to another matrix.
Band matrix
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal "band", comprising the main diagonal and zero or more diagonals on either side.
Tridiagonal matrix
A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. In particular, a tridiagonal matrix is a direct sum of "p" 1-by-1 and "q" 2-by-2 matrices such that "p" + "q"/2 = "n" — the dimension of the tridiagonal. Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have one of these properties. Furthermore, if a real tridiagonal matrix "A" satisfies "a" "a" > 0 for all "k", so that the signs of its entries are symmetric, then it is similar to a Hermitian matrix, by a diagonal change of basis matrix. Hence, its eigenvalues are real. If we replace the strict inequality by "a" "a" ≥ 0, then by continuity, the eigenvalues are still guaranteed to be real, but the matrix need no longer be similar to a Hermitian matrix.
Moore matrix
In linear algebra, a Moore matrix, introduced by , is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism applied to the first column, so it is an "m" × "n" matrix
Balanced matrix
In mathematics, a balanced matrix is a 0-1 matrix (a matrix where every entry is either zero or one) that does not contain any square submatrix of odd order having all row sums and all column sums equal to 2.
Definiteness of a matrix
In linear algebra, a symmetric real matrix formula_1 is said to be positive definite if the scalar formula_2 is strictly positive for every non-zero column vector formula_3 of formula_4 real numbers. Here formula_5 denotes the transpose of formula_3.
Nonnegative matrix
In mathematics, a nonnegative matrix, written
is a matrix in which all the elements are equal to or greater than zero, that is,
A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different.
Symmetry in mathematics
The integral of an even function from −"A" to +"A" is twice the integral from 0 to +"A" (where "A" is finite, and the function has no vertical asymptotes between −"A" and "A". This also holds true when "A" is infinite, but only if the integral converges).In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix "A" is symmetric if
Fundamental matrix (linear differential equation)
In mathematics, a fundamental matrix of a system of "n" homogeneous linear ordinary differential equations
Block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is "interpreted" as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.
Boolean matrix
In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the Boolean algebra has just two elements {0,1} the Boolean matrix is called a logical matrix.
Matrix unit
Let "I" be a nonempty set, to be used for counting the matrix rows and columns. There is no requirement for it to be finite; indeed, standard matrix algebra would use the set of natural numbers (not including zero) N. A matrix unit is either an ordered pair ("r", "c"), with "r" and "c" elements of "I", or it is a special "zero" object, written as "0". Multiplication is defined as follows:
The 0 element can be seen as an "error symbol" for when multiplication fails; the first rule implies that errors propagate through an entire product containing a single incompatible combination.
Matrix (mathematics)
The "determinant" det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in R) or volume (in R) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
Matrix unit
In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. The topic is comparatively obscure within linear algebra, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the context of abstract algebra, especially the theory of semigroups.
M-matrix
In mathematics, especially linear algebra, an "M"-matrix is a "Z"-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular "M"-matrices are a subset of the class of "P"-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name "M"-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.
Matrix (mathematics)
Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map "f": R → R. If "f", ..., "f" denote the components of "f", then the Jacobi matrix is defined as 
If "n" > "m", and if the rank of the Jacobi matrix attains its maximal value "m", "f" is locally invertible at that point, by the implicit function theorem.
Generalized eigenvector
The matrix
Matrix equivalence
In linear algebra, two rectangular "m"-by-"n" matrices "A" and "B" are called equivalent if
for some invertible "n"-by-"n" matrix "P" and some invertible "m"-by-"m" matrix "Q". Equivalent matrices represent the same linear transformation "V" → "W" under two different choices of a pair of bases of "V" and "W", with "P" and "Q" being the change of basis matrices in "V" and "W" respectively.
Quaternion algebra
In mathematics, a quaternion algebra over a field "F" is a central simple algebra "A" over "F" that has dimension 4 over "F". Every quaternion algebra becomes the matrix algebra by "extending scalars" (equivalently, tensoring with a field extension), i.e. for a suitable field extension "K" of "F", formula_1 is isomorphic to the 2×2 matrix algebra over "K".
Matrix (mathematics)
This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field, that is, a set where addition, subtraction, multiplication and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields. For example, coding theory makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (for example, to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset.
Arrowhead matrix
In the mathematical field of linear algebra, an arrowhead matrix is a square matrix containing zeros in all entries except for the first row, first column, and main diagonal, these entries can be any number. In other words, the matrix has the form
Matrix multiplication
In formula_56, the product is defined for every pair of matrices. This makes formula_56 a ring, which has the identity matrix as identity element (the matrix whose diagonal entries are equal to 1 and all other entries are 0). This ring is also an associative -algebra.
Minor (linear algebra)
Let A be an "m" × "n" matrix and "k" an integer with 0 < "k" ≤ "m", and "k" ≤ "n". A "k" × "k" minor of A, also called minor determinant of order k of A or, if formula_5, (n-k):th minor determinant of A, with the word "determinant" often omitted and the word "order" sometimes replaced by "degree", is the determinant of a "k" × "k" matrix obtained from A by deleting "m" − "k" rows and "n" − "k" columns. Sometimes the term is used to refer to the "k" × "k" matrix obtained from A as above (by deleting "m" − "k" rows and "n" − "k" columns), but this matrix should be referred to as a (square) submatrix of A, leaving the term "minor" to refer to the determinant of this matrix. For a matrix A as above, there are a total of formula_6 minors of size "k" × "k". Minor of order zero is often defined to be 1. For a square matrix, zeroth minor is just the determinant of the matrix.
Semisimple algebra
Any self-adjoint subalgebra "A" of "n" × "n" matrices with complex entries is semisimple. Let Rad("A") be the radical of "A". Suppose a matrix "M" is in Rad("A"). Then "M*M" lies in some nilpotent ideals of "A", therefore ("M*M")"" = 0 for some positive integer "k". By positive-semidefiniteness of "M*M", this implies "M*M" = 0. So "M x" is the zero vector for all "x", i.e. "M" = 0.
Unit fraction
Many well-known infinite series have terms that are unit fractions. These include:
The Hilbert matrix is the matrix with elements
It has the unusual property that all elements in its inverse matrix are integers. Similarly, defined a matrix with elements
where "F" denotes the "i"th Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.
Linear group
In mathematics, a matrix group is a group "G" consisting of invertible matrices over a specified field "K", with the operation of matrix multiplication, and a linear group is an abstract group that is isomorphic to a matrix group over a field "K", in other words, admitting a faithful, finite-dimensional representation over "K".
Transformation matrix
In linear algebra, linear transformations can be represented by matrices. If formula_1 is a linear transformation mapping formula_2 to formula_3 and formula_4 is a column vector with formula_5 entries, then