Search results
Results from the WOW.Com Content Network
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.
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]
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 ...
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 ...
The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A −T is sometimes used to represent either of these equivalent expressions.
Cramer's rule. In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one ...
In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix is an involution if and only if where is the identity matrix. Involutory matrices are all square roots of the identity matrix. This is a consequence of the fact that any invertible matrix multiplied by its inverse is the identity.
The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable multivariable function f : R n → R n is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible. In this case, the Jacobian of f −1 at f(p) is the matrix inverse of the ...