# [R] How to get the pseudo left inverse of a singular square

(Ted Harding) Ted.Harding at nessie.mcc.ac.uk
Thu Aug 14 21:41:01 CEST 2003

```On 14-Aug-03 Feng Zhang wrote:
> Thank, Jerome
>
> The question is if this generalized inverse can make
> their product to be identity matrix?

>> On August 14, 2003 09:24 am, Feng Zhang wrote:
>> > Dear R-listers,
>> >
>> > I have a dxr matrix Z, where d > r.
>> > And the product Z*Z' is a singular square matrix.
>> > The problem is how to get the left inverse U of this
>> > singular matrix Z*Z', such that U*(Z*Z') = I?

No, not if I is to be a full identity matrix (1s all the way
along the diagonal), when Z*Z' is singular. Compare ranks
on both sides; or simply observe that

det(U*(Z*Z')) = det(U)*det(Z*Z') = 0 while det(I) = 1.

However, you could have an incomplete diagonal (r 1s and
(n-r) 0s where r is the rank of Z*Z').

Ted.

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Date: 14-Aug-03                                       Time: 20:41:01
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