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The Octave programming language provides a pseudoinverse through the standard package function pinv and the pseudo_inverse() method. In Julia (programming language), the LinearAlgebra package of the standard library provides an implementation of the Moore–Penrose inverse pinv() implemented via singular-value decomposition. [25]
Instead of ([] []), we need to calculate directly or indirectly [citation needed] [original research? (), (), (), ().In a dense and small system, we can use singular value decomposition, QR decomposition, or Cholesky decomposition to replace the matrix inversions with numerical routines.
When the channel state information is perfectly known at the transmitter, the zero-forcing precoder is given by the pseudo-inverse of the channel matrix. Zero-forcing has been used in LTE mobile networks. [2]
So, a generalized inverse of () is also called a -constrained pseudoinverse of . An example of a pseudoinverse that can be used for the solution of a constrained problem is the Bott–Duffin inverse of constrained to , which is defined by the equation
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 a wider class of ...
A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.
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The adjoint and transpose of a pseudo-differential operator is a pseudo-differential operator. If a differential operator of order m is (uniformly) elliptic (of order m) and invertible, then its inverse is a pseudo-differential operator of order −m, and its symbol can be calculated. This means that one can solve linear elliptic differential ...