Let $V$ be a finite-dimensional Hilbert space and $Gr_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by extending $$ \langle v_1\wedge \cdots \wedge v_k, w_1\wedge \cdots \wedge w_k\rangle_{\bigwedge^k(V)} := \det{\left(\langle v_i,w_j\rangle_{V}\,_{i,j}\right)} $$ bilinearly to all $k$-vectors.

The Grassmannian can be equipped with a metric which measures the angle (more exactly I think the sine of the angle) between two $k$-dimensional subspaces by $$ d(U,W) := \left\| P_U-P_W \right\| $$ where $P_U:V\to U$ is the projection onto U and the norm is the operator norm (see for example this discussion and Equation (3), p. 3428 in the article linked therein).

I know that there is the Plücker embedding which maps $G_k(V)$ to the projectivisation of the $k$th exterior power. In fact, the image of the Plücker embedding corresponds to the simple $k$-vectors.

For example, if $U$ and $W$ coincide then the $k$-wedge of two distinct bases are off by a scalar factor (the determinant of the change of basis matrix). So it would be nice to have something like $$ d(U,W) \leq \left\|u_1\wedge\cdots\wedge u_k - w_1\wedge\cdots\wedge w_k\right\|_{\bigwedge^k(V)} $$ for any bases $(u_i)$ and $(w_i)$ of $U$ and $W$.

Is there a way of estimating the angle metric on the Grassmannian in terms of the norm on the exterior power?


$\newcommand{\R}{\mathbb{R}} \newcommand{\U}{\mathbf{U}} \newcommand{\W}{\mathbf{W}} \newcommand{\A}{\mathbf{A}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\rank}{\mathrm{rank}} $ Choosing an arbitrary basis, we may assume $V=\R^n$. We will show that the mapping \begin{align} \mathbb{S}\big(\bigwedge\nolimits^k_s(\R^n)\big)&\to Gr_k(\R^n)\\ \sigma=u_1\wedge\ldots\wedge u_k &\mapsto\operatorname{span}\{u_1,\ldots,u_k\}=\{v\in\R^n\colon v\wedge\sigma=0\} \end{align} is continuous w.r.t. the angle metric on the Grassmannian and induced inner product metric on the exterior power. The following lemma implies this assertion. Here $\mathbb{S}(X)$ denotes the set of elements in $X$ of unit length and $\bigwedge\nolimits^k_s(\R^n)$ is the set of simple $k$-vectors.

Let $U,W\in Gr_k(\R^n)$ and choose orthonormal bases $u_1,\ldots,u_k$ and $w_1,\ldots,w_k$ of $U$ and $W$, respectively. Let further $\U,\W\in\R^{n\times k}$ be the matrices with columns $(u_i)$ and $(w_i)$, respectively. Finally, define the unit $k$-vectors $\sigma=u_1\wedge\cdots\wedge u_k$ and $\tau=w_1\wedge\cdots\wedge w_k$.

Lemma \begin{align} d(U,W) \leq \sqrt{k}\left\|\sigma-\tau\right\|_{\bigwedge^k(\R^n)} \end{align}

Proof First, note that by the Cauchy-Schwartz inequality \begin{align} \det(\U^T\W)=\langle\sigma,\tau\rangle_{\bigwedge^k(\R^n)} \leq \|\sigma\|_{\bigwedge^k(\R^n)}\|\tau\|_{\bigwedge^k(\R^n)}=1. \tag{1}\label{eq:CS_ineq} \end{align} Secondly, the Euclidean operator norm of a linear operator $A$ is bounded by its Frobenius norm, i.e. $\|A\| \leq \|A\|_F$. This follows from the fact that the Euclidean norm and the Frobenius norm of a vector coincide and the submultiplicativity of the Frobenius norm. Therefore, \begin{align} d(U,W)^2 &= \left\| P_U-P_W \right\|^2 \leq \left\| P_U-P_W \right\|_F^2 =\left\| P_U\right\|_F^2 +\left\|P_W \right\|_F^2 -2\left\langle P_U,P_W \right\rangle_F. \end{align} Using that $P_U=\U^T\U$ and $P_W=\W^T\W$ are idempotent, we get that $\left\| P_U\right\|_F^2=\rank(P_U)=k$ and also $\left\| P_W\right\|_F^2=k$. Thus, \begin{align} \left\| P_U-P_W \right\|_F^2 &=2k-2\tr(\U\U^T\W\W^T) =k\big(2-\tfrac2k \tr((\U^T\W)^T\U^T\W)\big). \end{align} Due to the inequality of the arithmetic and geometric mean, it holds that \begin{align} \sqrt[k]{\det(\A)} \leq \tfrac1k \tr(\A) \end{align} for any matrix $\A\in\R^{k\times k}$ (because the trace is the sum and the determinant the product of the eigenvalues of $\A$). If $\det(\A)\leq1$ then for any $k\geq2$ \begin{align} \sqrt[2]{\det(\A)} \leq \tfrac1k \tr(\A). \tag{2}\label{eq:AGM} \end{align} Therefore we can estimate using \eqref{eq:CS_ineq} and \eqref{eq:AGM} for $\A=(\U^T\W)^T\U^T\W$ \begin{align} \left\| P_U-P_W \right\|_F^2 &\leq k\big(2-2\sqrt{\det((\U^T\W)^T\U^T\W)}\big) =k\big(2-2\det(\U^T\W)\big). \end{align} Note that we can always arrange for $\det(\U^T\W)$ to be positive be interchanging two columns of, say, $\U$. This is possible because we investigate the unoriented Grassmannian. Finally, direct computation shows that \begin{align} \|\sigma-\tau\|_{\bigwedge^k(\R^n)}^2=\|\sigma\|_{\bigwedge^k(\R^n)}^2+\|\tau\|_{\bigwedge^k(\R^n)}^2 - 2 \langle\sigma,\tau\rangle_{\bigwedge^k(\R^n)}^2 = 2-2\det(\U^T\W) \end{align} which concludes the proof.

The next statement is not really needed but just included here for completion.

Corollary For arbitrary $\sigma,\tau$ arbitrary $k$-vectors such that they span the subspaces $U$ and $W$, it holds that \begin{align} d(U,W) \leq \sqrt{k}\left\|\tfrac{\sigma}{\left\|\sigma\right\|_{\bigwedge^k(\R^n)}}- \tfrac{\tau}{\left\|\tau\right\|_{\bigwedge^k(\R^n)}}\right\|_{\bigwedge^k(\R^n)} \leq \sqrt{k}\frac{\left\|\sigma-\tau\right\|_{\bigwedge^k(\R^n)}}{\left\|\sigma\right\|_{\bigwedge^k(\R^n)}\left\|\tau\right\|_{\bigwedge^k(\R^n)}} \end{align}

Proof If $\sigma=a_1\wedge\cdots\wedge a_k$ for an arbitrary basis $(a_1,\ldots,a_k)$ of $U$ then we can change basis to an orthonormal basis $(u_1,\ldots,u_k)$, e.g. via QR-decomposition of $\A=\U\mathbf{R}$. Then $\sigma=\det(\mathbf{R})u_1\wedge\cdots\wedge u_k$ and $\left\|\sigma\right\|_{\bigwedge^k(\R^n)}=\det(\mathbf{R})$. Then use the lemma for $\sigma/\left\|\sigma\right\|_{\bigwedge^k(\R^n)}$ and $\tau/\left\|\tau\right\|_{\bigwedge^k(\R^n)}$. The second inequality follows from the fact that $2ab\leq a^2+b^2$ for any real numbers $a,b$.


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