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?
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What do you mean by a "nullspace matrix of $\mathbf A$"? – Misha Lavrov Apr 07 '18 at 16:24
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@Misha Lavrov By context, it is clear to me that the OP means a matrix which is a basis for the nullspace of A. – Mark L. Stone Apr 07 '18 at 20:07
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@MarkL.Stone I wanted to clarify this since I also noticed the orthonormality condition, which is not obvious to me from context. So, I wanted to know if there any additional conditions. – Misha Lavrov Apr 07 '18 at 20:32
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@Misha Lavro I addressed that in my answer below. – Mark L. Stone Apr 07 '18 at 20:40
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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, $W^HW = I_{m-n}$.
$\|{\bf W}{\bf W}^H-{\bf I}_m\|_F^2 = trace((WW^H - I_m)(WW^H - I_m)^H) = trace(W(W^HW)W^H - WW^H - WW^H + I_m) = trace(-WW^H + I_m) = -trace(W^HW) + trace(I_m) = -(m - n) + m = n$
Mark L. Stone
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