Category: Matrix theory

Matrix determinant lemma
In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row ve
Totally positive matrix
In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. A totally positive matrix ha
Rouché–Capelli theorem
In linear algebra, the Rouché–Capelli theorem determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is various
Loewner order
In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/c
Matrix semialgebra
No description available.
In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(M,q) has a solution for every vector q.
Crouzeix's conjecture
Crouzeix's conjecture is an unsolved (as of 2018) problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it refines Crouzeix's theorem, which states: where the set is the field of
Minimal polynomial (linear algebra)
In linear algebra, the minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynom
Trigonometric functions of matrices
The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor s
Sylvester's formula
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial
Frobenius inner product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted . The operation is a component-wise inner product of two matrices as
GCD matrix
In mathematics, a (sometimes abbreviated as GCD matrix) is a matrix/
Generalized eigenvector
In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector sp
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 s
Companion matrix
In linear algebra, the Frobenius companion matrix of the monic polynomial is the square matrix defined as . Some authors use the transpose of this matrix, which (dually) cycles coordinates, and is mor
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 ma
Sherman–Morrison formula
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix and the outer pro
In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent matrix if
Computing the permanent
In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of
Sylvester's criterion
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester. Sylvester's criter
Change of basis
In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called co
SMAWK algorithm
The SMAWK algorithm is an algorithm for finding the minimum value in each row of an implicitly-defined totally monotone matrix. It is named after the initials of its five inventors, Peter Shor, Shlomo
Commuting matrices
In linear algebra, two matrices and are said to commute if , or equivalently if their commutator is zero. A set of matrices is said to commute if they commute pairwise, meaning that every pair of matr
Gershgorin circle theorem
In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's
Specht's theorem
In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940. Two m
Matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matr
Eigendecomposition of a matrix
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matric
Khatri-Rao product
No description available.
Whitehead's lemma
Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form is equivalent to the identity matrix by elementary transformations (that is,
Spectral abscissa
In mathematics, the spectral abscissa of a matrix or a bounded linear operator is the greatest real part of the matrix's spectrum (its set of eigenvalues). It is sometimes denoted . As a transformatio
Sinkhorn's theorem
Sinkhorn's theorem states that every square matrix with positive entries can be written in a certain standard form.
Nullity theorem
The nullity theorem is a mathematical theorem about the inverse of a partitioned matrix, which states that the nullity of a block in a matrix equals the nullity of the complementary block in its inver
Trace (linear algebra)
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for
Matrix semiring
No description available.
Eigenvalues and eigenvectors of the second derivative
Explicit formulas for eigenvalues and eigenvectors of the second derivative with different boundary conditions are provided both for the continuous and discrete cases. In the discrete case, the standa
Mutual coherence (linear algebra)
In linear algebra, the coherence or mutual coherence of a matrix A is defined as the maximum absolute value of the cross-correlations between the columns of A. Formally, let be the columns of the matr
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if A is the
Jordan matrix
In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the zero 0 and one 1), where each block alo
Manin matrix
In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave
Bendixson's inequality
In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary parts of Characteristic roots (e
Jacobi's formula
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. If A is a differentiable map from the real numbers
Smith normal form
In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith
Weinstein–Aronszajn identity
In mathematics, the Weinstein–Aronszajn identity states that if and are matrices of size m × n and n × m respectively (either or both of which may be infinite) then,provided (and hence, also ) is of t
Analytic function of a matrix
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exp
Poincaré separation theorem
In mathematics, the Poincaré separation theorem gives the upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real sym
Matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries in R is a
In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of -matrices, which are the closure of the class of P-matrices, with every
Adjugate matrix
In linear algebra, the adjugate or classical adjoint of a square matrix A is the transpose of its cofactor matrix and is denoted by adj(A). It is also occasionally known as adjunct matrix, or "adjoint
Bidiagonalization is one of unitary (orthogonal) matrix decompositions such that U* A V = B, where U and V are unitary (orthogonal) matrices; * denotes Hermitian transpose; and B is upper bidiagonal.
Moore–Penrose inverse
In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in
Polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix, both square and
Partial inverse of a matrix
In linear algebra and statistics, the partial inverse of a matrix is an operation related to Gaussian elimination which has applications in numerical analysis and statistics. It is also known by vario
Schur–Horn theorem
In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired i
Sparse graph code
A Sparse graph code is a code which is represented by a sparse graph. Any linear code can be represented as a graph, where there are two sets of nodes - a set representing the transmitted bits and ano
Schur decomposition
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square
Eigenvalues and eigenvectors
In linear algebra, an eigenvector (/ˈaɪɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is a
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is deno
Logarithmic norm
In mathematics, the logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operator norm. The logarithmic norm was indepe
Computational complexity of matrix multiplication
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms ar
Amitsur–Levitzki theorem
In algebra, the Amitsur–Levitzki theorem states that the algebra of n × n matrices over a commutative ring satisfies a certain identity of degree 2n. It was proved by Amitsur and Levitsky. In particul
Operator monotone function
In linear algebra, the operator monotone function is an important type of real-valued function, first described by Charles Löwner in 1934. It is closely allied to the operator concave and operator con
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matri
Matrix multiplication algorithm
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multipl
Numerical range
In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex matrix A is the set where denotes the conjugate transpose of the vector . The numer
Spread of a matrix
In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix.
Weighing matrix
In mathematics, a weighing matrix of order and weight is a matrix with entries from the set such that: Where is the transpose of and is the identity matrix of order . The weight is also called the deg
Productive matrix
In linear algebra, a square nonnegative matrix of order is said to be productive, or to be a Leontief matrix, if there exists a nonnegative column matrix such as is a positive matrix.
Lie product formula
In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, named after Hale Trotter, states that for arbitrary m × m real or complex matr
Perron–Frobenius theorem
In matrix theory, the Perron–Frobenius theorem, proved by Oskar Perron and Georg Frobenius, asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the co
Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositio
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In
Matrix sign function
In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function. It was introduced by J.D. Roberts in 1971 as a tool for model reduction and for
Minor (linear algebra)
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows and columns. Minors obtained by removing just one row and
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis
Faddeev–LeVerrier algorithm
In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial of a square matrix, A, named after Dmitry Konstant
Block matrix pseudoinverse
In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal p
Square root of a matrix
In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A. Some authors
Cuthill–McKee algorithm
In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee, is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a ba
Spectrum of a matrix
In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars such that is
Tracy-Singh product
No description available.
Matrix completion
Matrix completion is the task of filling in the missing entries of a partially observed matrix, which is equivalent to performing data imputation in statistics. A wide range of datasets are naturally
Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n matrix B as a weighted sum of minors, which
Trace diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finit
Workshop on Numerical Ranges and Numerical Radii
Workshop on Numerical Ranges and Numerical Radii (WONRA) is a biennial workshop series on numerical ranges and numerical radii which began in 1992.
Frobenius covariant
In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices Ai associated with the eigenvalues and eigenvectors of A. They are named after
Sylvester's determinant identity
In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof
Haynsworth inertia additivity formula
In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix a
International Linear Algebra Society
The International Linear Algebra Society (ILAS) is a professional mathematical society organized to promote research and education in linear algebra, matrix theory and matrix computation. It serves th
Derivative of the exponential map
In the theory of Lie groups, the exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map reduces to the matrix exponential. The ex
Permanent (mathematics)
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Bot
In mathematics, an eigenoperator, A, of a matrix H is a linear operator such that where is a corresponding scalar called an eigenvalue.
Freivalds' algorithm
Freivalds' algorithm (named after Rūsiņš Mārtiņš Freivalds) is a probabilistic randomized algorithm used to verify matrix multiplication. Given three n × n matrices , , and , a general problem is to v
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theo
Matrix calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with re
Orthogonal Procrustes problem
The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices and and asked to find an orthogonal matrix which most closely ma
Weyr canonical form
In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix satisfying certain conditions. A square matrix is said to be in the Weyr canonical form if th
Frobenius determinant theorem
In mathematics, the Frobenius determinant theorem was a conjecture made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in, with an English tr
Kronecker sum of discrete Laplacians
In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid domai
Hadamard product (matrices)
In mathematics, the Hadamard product (also known as the element-wise product, entrywise product or Schur product) is a binary operation that takes two matrices of the same dimensions and produces anot
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 V is factorized into (usuall
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or co
Trace inequality
In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matric
In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows: In general, there are n2 quasideterm
Carleman matrix
In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which
Woodbury matrix identity
In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k c
Antieigenvalue theory
In applied mathematics, antieigenvalue theory was developed by from 1966 to 1968. The theory is applicable to numerical analysis, wavelets, statistics, quantum mechanics, finance and optimization. The
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent. Let be a partition of an in
Khatri–Rao product
In mathematics, the Khatri–Rao product of matrices defined as in which the ij-th block is the mipi × njqj sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and
Golden–Thompson inequality
In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the con
Minimum degree algorithm
In numerical analysis, the minimum degree algorithm is an algorithm used to permute the rows and columns of a symmetric sparse matrix before applying the Cholesky decomposition, to reduce the number o
Schur product theorem
In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix.The result is named afte
Spark (mathematics)
In mathematics, more specifically in linear algebra, the spark of a matrix is the smallest integer such that there exists a set of columns in which are linearly dependent. If all the columns are linea
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 where In denotes the n-by-n identity matrix a
In astronomical and geodetic calculations, Cracovians are a clerical convenience introduced in the 1930s by Tadeusz Banachiewicz for solving systems of linear equations by hand. Such systems can be wr