an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can...

and in particular linear algebra, the Moore–Penrose inverse ⁠ A + {\displaystyle A^{+}} ⁠ of a matrix ⁠ A {\displaystyle A} ⁠, often called the pseudoinverse...

{\displaystyle n\times n} identity matrix. The matrix Ω {\displaystyle \Omega } has determinant + 1 {\displaystyle +1} and its inverse is Ω − 1 = Ω T = − Ω {\displaystyle...

left inverse of a square matrix is also a right-inverse (see Invertible matrix theorem). For other proofs, see below. Let A be an n × n matrix with entries...

entries), an invertible matrix is a matrix that has an inverse that is also an integer matrix. Such a matrix is called a unimodular matrix for distinguishing...

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...

unknowns are different so that matrix F {\displaystyle F} is not square. However, even a square matrix can have no inverse: matrix F {\displaystyle F} can be...

Inverse element Inverse function, a function that "reverses" another function Generalized inverse, a matrix that has some properties of the inverse matrix...

computes the inverse of a "rank-1 update" to a matrix whose inverse has previously been computed. That is, given an invertible matrix A {\displaystyle...

may be satisfactory. The inverse iteration algorithm requires solving a linear system or calculation of the inverse matrix. For non-structured matrices...

is unique and is called the inverse matrix of A {\displaystyle A} , denoted A − 1 {\displaystyle A^{-1}} . A square matrix A {\displaystyle A} that is...

to the identity matrix I {\displaystyle \mathbf {I} } , the right-hand n × n {\displaystyle n\times n} block is then the inverse matrix A − 1 {\displaystyle...

formulas for the inverse matrix V − 1 {\displaystyle V^{-1}} . In particular, Lagrange interpolation shows that the columns of the inverse matrix V − 1 = [ 1...

the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function f : Rn → Rn is the Jacobian matrix of the inverse function...

Q^{\mathrm {T} }=Q^{-1},} where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where...

The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices...

multiplicative inverse. For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix A...

algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors...

transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A−T is sometimes...

where A−1 is the inverse matrix of A. If A has no inverse, solutions—if any—can be found using its generalized inverse. Matrices and matrix multiplication...

corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an...

In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U...

prior for the covariance matrix of a multivariate normal distribution. We say X {\displaystyle \mathbf {X} } follows an inverse Wishart distribution, denoted...

calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices. The requirement that the square matrix be smaller...

,} where I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its...

its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for holomorphic functions,...

In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. It is a more...

matrix-related notions is about properties of products or inverses of the given matrix. The matrix product of a m-by-n matrix A and a n-by-k matrix B...

the Hadamard product and 1 is a constant vector with elements 1. The inverse matrix-to-vector diag operator is sometimes denoted by the identically named...

multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements x, y such that xy = 0. A square matrix has an inverse if and only...