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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]
The explicit inverse of a Hermitian matrix can be computed by Cholesky decomposition, in a manner similar to solving linear systems, using operations (multiplications). [10] The entire inversion can even be efficiently performed in-place.
Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. It is also sometimes referred to as LR decomposition (factors into left and right triangular matrices).
Calculating the inverse matrix once, and storing it to apply at each iteration is of complexity O(n 3) + k O(n 2). Storing an LU decomposition of () and using forward and back substitution to solve the system of equations at each iteration is also of complexity O(n 3) + k O(n 2).
In mathematics, and in particular linear algebra, the Moore–Penrose inverse + of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. [1] It was independently described by E. H. Moore in 1920, [2] Arne Bjerhammar in 1951, [3] and Roger Penrose in 1955. [4]
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems () = and = require fewer additions and multiplications to solve, compared with the original system =, though one might require significantly more digits in inexact arithmetic such as floating point.
Gaussian elimination is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix is first created with the left side being the matrix to invert and the right side being the identity matrix. Then, Gaussian elimination is used to convert the left side into the identity matrix ...