Frobenius norm property

Question

Say $A$ be an $m \times m$ matrix and $x$ be a nonzero vector of $m \times 1$. I've encountered the following equality:

$$ {\left \| A \left( I - \frac{x x^H}{x^H x} \right) \right\|}^2_{F} = {\|A\|}^2_{F} - \frac{{\|Ax\|}^2_{2}}{x^H x}$$ where $\|.\|_{F}$ is the Frobenius norm. Can anybody provide the intermediate steps leading to the right side of the equation. Any help would be appreciated.

Answer 1

• Expand the left hand side using $\| M \|_{F}^2 = tr( M^H M )$
• Recall that $tr(AB) = tr(BA)$ to reorganize the terms with $A^H A x x^H$