Frobenius norm of $||AA^+ – I||_F = ? $

Question

I need to find a value for the following norm $||AA^+ – I||_F$, where:

• $A^+$ is the Moore–Penrose Inverse matrix
• $||A||_F = \sqrt{Tr(AA^T)}$
• A have $n \times m$ dimension and have rank $r$

I have try to do the SVD decomposition of the matrix $AA^+ – I$, but I couldn't go further because I don't know how the decomposition in SVD of $A$ have a relation with $A^+$.

Answer 1

Over $\mathbb{R}$.

$AA^+$ is a symmetric matrix, the eigenvalues of which, being $1$ ($r\;\times$) and $0$ ($n-r\;\times$).

Then $AA^+-I_n$ is orthogonally similar to $D=diag(0_r,-I_{n-r})$.

Finally $||AA^+-I_n||_F=||D||_F=\sqrt{n-r}$.