Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition pp 55-86 | Cite as

# Generalized Inverse Matrices

## Abstract

Let * A* be a square matrix of order

*n*. If it is nonsingular, then Ker(

*) = {*

**A****0**} and, as mentioned earlier, the solution vector

*in the equation*

**x***=*

**y***is determined uniquely as*

**Ax***=*

**x**

**A**^{-1}

*. Here,*

**y**

**A**^{-1}is called the inverse (matrix) of

*defining the inverse transformation from*

**A***∈ E*

**y**^{n}to

*∈ E*

**x**^{m}, whereas the matrix

*represents a transformation from*

**A***to*

**x***. When*

**y***is*

**A***n*by

*m*,

*=*

**Ax***has a solution if and only if*

**y***∈ Sp(*

**y***). Even then, if Ker(*

**A***) ≠ {*

**A***}, there are many solutions to the equation*

**A***=*

**Ax***due to the existence of*

**A**

**x**_{0}(≠ 0) such that

**Ax**_{0}=

**0**, so that

*(*

**A***+*

**x**

**x**_{0}) =

*. If*

**y***∉ Sp(*

**y***), there is no solution vector to the equation*

**A***=*

**Ax***.*

**y**### Keywords

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