invertible matrix (non-singular, non-degenarate or regular) is a square matrix that has an inverse. In other words, if some other matrix is multiplied...

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

determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally...

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...

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...

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

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...

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...

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...

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...

{\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...

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...

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...

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...

inverse function. The inverse function is also differentiable, and the inverse function rule expresses its derivative as the multiplicative inverse of...

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...

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...

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...

matrix is called non-singular (or invertible) when it has an inverse; by definition, a matrix that fails this criterion is singular. In more algebraic terms...

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

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...

The inverse of a Frobenius matrix is again a Frobenius matrix, equal to the original matrix with changed signs outside the main diagonal. The inverse of...

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

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...

post-multiplying an n-column matrix M, forming MP, permutes the columns of M. Every permutation matrix P is orthogonal, with its inverse equal to its transpose:...

,} 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...

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

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...