Search results
Results From The WOW.Com Content Network
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. [19]
A square matrix may have a multiplicative inverse, called an inverse matrix. In the common case where the entries belong to a commutative ring R, a matrix has an inverse if and only if its determinant has a multiplicative inverse in R. The determinant of a product of square matrices is the product of the determinants of the factors.
Matrix formulae to calculate rows and columns of LU factors by recursion are given in the remaining part of Banachiewicz's paper as Eq. (2.3) and (2.4) (see F90 code example). This paper by Banachiewicz contains both derivation of and factors of respectively non-symmetric and symmetric matrices. They are sometimes confused as later publications ...
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I]
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
I is the 3 × 3 identity matrix (which is trivially involutory); R is the 3 × 3 identity matrix with a pair of interchanged rows; S is a signature matrix. Any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.
A square matrix having a multiplicative inverse, that is, a matrix B such that AB = BA = I. Invertible matrices form the general linear group. Involutory matrix: A square matrix which is its own inverse, i.e., AA = I. Signature matrices, Householder matrices (Also known as 'reflection matrices' to reflect a point about a plane or line) have ...
Any real symplectic matrix can be decomposed as a product of three matrices: = ′, where and ′ are both symplectic and orthogonal, and is positive-definite and diagonal. [6] This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.