[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.
[2] Over a field, a square matrix that is not invertible is called singular or degenerate.
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.
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).
This is true because singular matrices are the roots of the determinant function.
Gaussian elimination is a useful and easy way to compute the inverse of a matrix.
The reason it works is that the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using elementary matrices (
we create the augumented matrix by combining A with I and applying Gaussian elimination.
A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed: Victor Pan and John Reif have done work that includes ways of generating a starting seed.
[5][6] Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough.
Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic.
satisfying the linear Diophantine equation The formula can be rewritten in terms of complete Bell polynomials of arguments
Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate: If matrix A is positive definite, then its inverse can be obtained as where L is the lower triangular Cholesky decomposition of A, and L* denotes the conjugate transpose of L. Writing the transpose of the matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices, but this recursive method is inefficient for large matrices.
) is invertible, its inverse is given by The determinant of A, det(A), is equal to the triple product of x0, x1, and x2—the volume of the parallelepiped formed by the rows or columns: The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide.
Intuitively, because of the cross products, each row of A–1 is orthogonal to the non-corresponding two columns of A (causing the off-diagonal terms of
For example, the first diagonal is: With increasing dimension, expressions for the inverse of A get complicated.
For n = 4, the Cayley–Hamilton method leads to an expression that is still tractable: Matrices can also be inverted blockwise by using the following analytic inversion formula:[9] where A, B, C and D are matrix sub-blocks of arbitrary size.
[10]) This strategy is particularly advantageous if A is diagonal and D − CA−1B (the Schur complement of A) is a small matrix, since they are the only matrices requiring inversion.
This technique was reinvented several times and is due to Hans Boltz (1923),[citation needed] who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness.
satisfies the invertibility condition for its left upper block A.
These formulas together allow to construct a divide and conquer algorithm that uses blockwise inversion of associated symmetric matrices to invert a matrix with the same time complexity as the matrix multiplication algorithm that is used internally.
[12] Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of O(n2.371552) operations, while the best proven lower bound is Ω(n2 log n).
As such, it satisfies Therefore, only 2L − 2 matrix multiplications are needed to compute 2L terms of the sum.
More generally, if A is "near" the invertible matrix X in the sense that then A is nonsingular and its inverse is If it is also the case that A − X has rank 1 then this simplifies to If A is a matrix with integer or rational entries and we seek a solution in arbitrary-precision rationals, then a p-adic approximation method converges to an exact solution in O(n4 log2 n), assuming standard O(n3) matrix multiplication is used.
[15] The method relies on solving n linear systems via Dixon's method of p-adic approximation (each in O(n3 log2 n)) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.
Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns).
However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.
Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations.
Matrix inversion also plays a significant role in the MIMO (Multiple-Input, Multiple-Output) technology in wireless communications.
The signal arriving at each receive antenna will be a linear combination of the N transmitted signals forming an N × M transmission matrix H. It is crucial for the matrix H to be invertible for the receiver to be able to figure out the transmitted information.