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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 entries are given by the divisor function; entires of the inverse are given by the Möbius function. a ij are 1 if i divides j or if j = 1; otherwise, a ij = 0. A (0, 1)-matrix. Shift matrix: A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. a ij = δ i+1,j or a ij = δ i−1,j
According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function f : R n → R n is the Jacobian matrix of the inverse function. That is, the Jacobian matrix of the inverse function at a point p is = (()), and the Jacobian determinant is
Inverse-Wishart. In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.
Matrix equivalence is an equivalence relation on the space of rectangular matrices. For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions. The matrices can be transformed into one another by a combination of elementary row and column operations. Two matrices are equivalent if and ...
Givens rotation. In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory.
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, below ...
The inverse of a matrix has each eigenvalue inverted. A uniform scaling matrix is analogous to a constant number. In particular, the zero is analogous to 0, and; the identity matrix is analogous to 1. An idempotent matrix is an orthogonal projection with each eigenvalue either 0 or 1. A normal involution has eigenvalues .