Multiplication of matrices ${\bf W}$ and ${\bf W}^H$, which are the span of nullspace

Question

When ${\bf W}\in \mathcal{C}^{m\times (m-n)}$ is a nullspace matrix of ${\bf A}\in\mathcal{C}^{n\times m}$, where $m>n$, I have confirmed that ${\bf W}^H{\bf W}={\bf I}_{m-n}$. Here, ${\bf I}_{m-n}$ is an ($m-n$)-dimensional identity matrix. Now, I am trying to find any property for ${\bf W}{\bf W}^H \in\mathcal{C}^{m\times m}$, which is not an identity matrix. For fixed $n$, as $m\rightarrow\infty$, is it true that ${\bf W}{\bf W}^H\rightarrow {\bf I}_m$ ? I have observed that $\|{\bf W}{\bf W}^H-{\bf I}_m\|_F^2=n$ regardless of $m$. Anyone can provide any property related this?

Answer 1

This is true providing W is an orthonormal basis for the nullsapce. Such basis will always exist. Perhaps you have been using MATLAB's null command, or something similar, which always produces an orthonormal basis.

By definition of othornormality, $W^HW = I_{m-n}$.

$\|{\bf W}{\bf W}^H-{\bf I}_m\|_F^2 = trace((WW^H - I_m)(WW^H - I_m)^H) = trace(W(W^HW)W^H - WW^H - WW^H + I_m) = trace(-WW^H + I_m) = -trace(W^HW) + trace(I_m) = -(m - n) + m = n$