# Hello,The pseudoinverse formula for a matrix A is given by:P =

## Main Question or Discussion Point

Hello,

The pseudoinverse formula for a matrix A is given by:

P = (A$^{T}$A)$^{-1}$A$^{T}$

I remember knowing this some time ago and this has me worried now....why is (A$^{T}$A)$^{-1}$ guaranteed to exist? I know it is a square matrix but it could still be degenerate, right?

Would appreciate any help.

Thanks,
Luca

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Deveno

it's not guaranteed to exist.

for example, consider A =

[1 1]
[0 0].

the (left) pseudoinverse is useful when you have an nxk matrix (n > k) of rank k. then ATA is an invertible kxk matrix, so

(ATA)-1AT acts as a left-inverse (to the left-identity) for A, as you can easily verify by computation.

(in this case, In is merely a left-identity for the nxk matrices, the right-identity is Ik, and these matrices are of different sizes).

in other words, the pseudo-inverse "acts" like an inverse on a class of matrices which don't have inverses.