# [R] How to get the pseudo left inverse of a singular squarem atrix?

alka seltzer int20h at yahoo.com
Fri Feb 6 16:47:15 CET 2004

```>I'm rusty, but not *that* rusty here, I hope.
>
>If W (=Z*Z' in your case) is singular, it can not
have >inverse, which by
>definition also mean that nothing multiply by it will
>produce the identity
>matrix (for otherwise it would have an inverse and
>thus nonsingular).
>
>The definition of a generalized inverse is something
>like:  If A is a
>non-null matrix, and G satisfy AGA = A, then G is
>called a generalized
>inverse of A.  This is not unique, but a unique one
>that satisfy some
>additional properties is the Moore-Penrose inverse.
I >don't know if this is
>what ginv() in MASS returns, as I have not used it
>before.

Andy

The inverse of a Matrix A is defined as a Matrix B
such that B*A=A*B=I and not just B*A=I. But there are
matrices B for singular matrices A such that B*A=I but
A*B != I, therefore there exist "left-inverses" (or
"right-inverses") for non-invertable matrices.

Best Regards

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