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linalg::pseudoInverse -- Moore-Penrose inverse of a matrix

Introduction

linalg::pseudoInverse(A) computes the Moore-Penrose inverse of A.

Call(s)

linalg::pseudoInverse(A)

Parameters

A - a matrix of category Cat::Matrix

Returns

a matrix of the same domain type as A, or the value FAIL.

Related Functions

_invert

Details

Example 1

The Moore-Penrose inverse of the 2 x 3 matrix:

>> A := Dom::Matrix(Dom::Complex)([[1, I, 3], [1, 3, 2]])
                               +-         -+
                               |  1, I, 3  |
                               |           |
                               |  1, 3, 2  |
                               +-         -+

is the 3 x 2 matrix:

>> Astar := linalg::pseudoInverse(A)
                   +-                                -+
                   |   7/96 + 1/32 I,  1/24 - 1/32 I  |
                   |                                  |
                   |  - 7/32 - 5/96 I, 5/16 + 7/96 I  |
                   |                                  |
                   |   7/24 + 1/16 I,  1/96 - 3/32 I  |
                   +-                                -+ 

Note that in this example, only:

>> A * Astar
                                +-      -+
                                |  1, 0  |
                                |        |
                                |  0, 1  |
                                +-      -+

yields the identity matrix, but not (see ``Backgrounds'' below):

>> Astar * A
          +-                                                  -+
          |       11/96,      3/32 - 1/48 I,   29/96 + 1/32 I  |
          |                                                    |
          |   3/32 + 1/48 I,      95/96,      - 1/32 - 1/96 I  |
          |                                                    |
          |  29/96 - 1/32 I, - 1/32 + 1/96 I,      43/48       |
          +-                                                  -+

Background

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