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Sherman–Morrison formula. In linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of a " rank -1 update" to a matrix whose inverse has previously been computed. [1][2][3] That is, given an invertible matrix and the outer product of vectors and the formula cheaply computes an ...
Moore–Penrose inverse. 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]
Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] 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.
Woodbury matrix identity. In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, [1][2] says that the inverse of a rank- k correction of some matrix can be computed by doing a rank- k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion ...
Matrix (mathematics) An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix (pl.: matrices) is a rectangular array or table of ...
Toeplitz matrix. In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: Any matrix of the form. is a Toeplitz matrix. If the element of is denoted then we have.
An m × n rectangular Vandermonde matrix such that m ≤ n has rank m if and only if all x i are distinct. An m × n rectangular Vandermonde matrix such that m ≥ n has rank n if and only if there are n of the x i that are distinct. A square Vandermonde matrix is invertible if and only if the x i are distinct. An explicit formula for the ...
Generalized inverse. In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. 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 ...
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