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In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or —rarely used— regular) if there exists an n-by-n square matrix B such that
where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.[1] If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the inverse matrix of an invertible matrix.[citation needed]
Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices, i.e. m-by-n matrices for which m ≠ n, do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, (n ≤ m), then A has a left inverse, an n-by-m matrix B such that BA = In. If A has rank m (m ≤ n), then it has a right inverse, an n-by-m matrix B such that AB = Im.
While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings). However, in the case of a ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group, the general linear group of degree n, denoted GLn(R).
^Axler, Sheldon (18 December 2014). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer Publishing (published 2015). p. 296. ISBN 978-3-319-11079-0.
n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or —rarely used— regular) if there exists an n-by-n square matrix B such that...
general linear group of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero). Over the real...
invertible matrices is again invertible, and the inverse of an invertiblematrix is invertible, with the identity matrix as the identity element of the...
,} where I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertiblematrix can be found by dividing its...
matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix...
an invertiblematrix B {\displaystyle B} with conjugate transpose B ∗ {\displaystyle B^{*}} such that M = B ∗ B . {\displaystyle M=B^{*}B.} A matrix is...
identity matrix. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any invertiblematrix multiplied...
m\times n} matrix A {\displaystyle A} . A square matrix A {\displaystyle A} is called invertible or non-singular if there exists a matrix B {\displaystyle...
{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 Q∗ is the...
in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertiblematrix A and the dyadic product, u vT, of...
of 2 matrices, and for sets of matrices which contain at most one invertiblematrix. Paterson, Michael S. (1970). "Unsolvability in 3 × 3 matrices". Studies...
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...
Neumann series can be used for approximate matrix inversion. To approximate the inverse of an invertiblematrix A {\displaystyle \mathbf {A} } , we can assign...
\dots ,\ x_{n}} are distinct, then V is a square matrix with non-zero determinant, i.e. an invertiblematrix. Thus, given V and y, one can find the required...
function is locally invertible near this point, that is, there is a neighbourhood of this point in which the function is invertible. The (unproved) Jacobian...
The transpose of an invertiblematrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A−T is...
=\mathbf {I} .} A matrix that has an inverse is an invertiblematrix. Otherwise, it is a singular matrix. A product of matrices is invertible if and only if...
of a Lie algebra, when the Lie algebra is gln Invertiblematrix (this usage is rare) QS Regular Matrix, a quadraphonic sound system developed by Sansui...
congruent if there exists an invertiblematrix P over the same field such that PTAP = B where "T" denotes the matrix transpose. Matrix congruence is an equivalence...
the inverse of an invertiblematrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence...
invertible matrices. In fact, this map is surjective which means that every invertiblematrix can be written as the exponential of some other matrix (for...
an invertible element is an element that has an inverse. In a ring, an invertible element, also called a unit, is an element that is invertible under...
capacitance matrix. Sherman–Morrison formula Schur complement Matrix determinant lemma, formula for a rank-k update to a determinant Invertiblematrix Moore–Penrose...
the left by an n × n invertiblematrix S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the same size according...
diagonal matrix is invertible if and only if each of its main-diagonal blocks are invertible, and in this case its inverse is another block diagonal matrix given...
represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map...
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