I am reading through this paper by Stewart on oblique projectors, i.e. matrices $P \in \mathbb{C}^{n \times n}$ where $P^2 = P$.
He describes a canonical form for projectors as follows.
Let $P$ be be a projector of order $n$ and rank $p$. Then there is an orthogonal matrix $Q$ such that
$$
QPQ^* =
\begin{bmatrix}
I_p & T \\
0 & 0
\end{bmatrix},
$$ where $T$ is a $p \times (n-p)$ diagonal matrix whose entries are the tangents of the canonical angles between the column space of P and the row space of P.
The proof for this form can be found in this paper by Dokovic.
However, I am unable to figure out why $T$ contains the tangents of the canonical angles.
From an initial reading of the proof, it appears that the entries of $T$ ($\sigma_i$ in the paper) are the inner product of an eigenvector $v$ of $PP^*$ and a perpendicular vector from the subspace spanned by $\{v, P^*v\}$.
Can someone show me how this corresponds to the tangents of the canonical angles? Or, point me to a different approach to get this result.
Best Answer
These principal angles are invariant under unitary changes of basis. Thus, it suffices to show that this holds for the canonical-form projection $\tilde P = QPQ^*$.
The row-space $\mathcal V$ and column space $\mathcal W$ of $\tilde P$ are the column spaces of the matrices $$ V = \pmatrix{I_p\\0} \quad W = \pmatrix{I_p\\T} $$ respectively. Following your characterization of principal angles, we can proceed as follows. The first (largest) principal angle corresponds to the largest $\cos \theta$ for the angle obtained between a vector $w \in \mathcal W$ and the subspace $\mathcal V$. That is, it suffices to maximize $\frac{\|\mathrm{proj}_{\mathcal V}(w)\|}{\|w\|}$ for $w \in \mathcal W$. For $w = Wx$, $\operatorname{proj}_\mathcal V(w) = x$. Thus, our aim is to maximize $\frac{\|x\|}{\|Wx\|}$, which means that we must minimize $\frac{\|Wx\|}{\|x\|}$, which (noting that $W^*W$ is diagonal) occurs when $x = e_p = (0,\dots,0,1) \in \Bbb C^p$ (assuming that the $\sigma_j$ are in non-increasing order).
Thus, we conclude that the principal vectors associated with the principal angle are the final column of $V$, $v = e_p \in \Bbb C^n$ and the final column of $W$, $w = e_p + \sigma_p e_{n} \in \Bbb C^n$. Now, verify that the tangent of the angle between these vectors is $\sigma_n$.
Finding the next principal angle and components amounts to considering the same setup as the above, only with the final columns of $V$ and $W$ removed. By following the same process, we see that $\sigma_{n-1}$ is the principal angles between these smaller subspaces, and the pattern continues for the remaining $\sigma_j$.