Lab 17: Grassmannians and Distances Between Subspaces

Lab goals

In this lab you will compute and interpret distances between subspaces. The main computational objects are:

orthonormal basis matrices $Q_U$ and $Q_W$;
the matrix $Q_U^TQ_W$;
singular values $\sigma_i$;
principal angles $\theta_i=\cos^{-1}(\sigma_i)$;
Grassmannian distances such as geodesic and chordal distance.

Python practice notebook

The Jupyter notebook version contains guided computations, solutions, and similar practice questions.

Download Lab 17 notebook

Google Colab link:

Open in Google Colab

Interactive lab

Use the interactive HTML page to visualize principal angles between two planes in $\mathbb R^3$ and to compute Grassmannian distances.

Open Lab 17 Interactive

Background formulas

If $Q_U$ and $Q_W$ are orthonormal basis matrices for two $k$-dimensional subspaces, then the principal angles are computed by

\[ Q_U^TQ_W=A_0\Sigma B_0^T, \qquad \cos\theta_i=\sigma_i. \]

The Grassmann geodesic distance is

\[ d_{\operatorname{Gr}}(U,W)=\left(\sum_{i=1}^k\theta_i^2\right)^{1/2}. \]

The chordal distance is

\[ d_{\mathrm{chordal}}(U,W)=\left(\sum_{i=1}^k\sin^2\theta_i\right)^{1/2}. \]

The projection representation is

\[ P_U=Q_UQ_U^T. \]

Independent-study tasks

Task 1: Two lines

Let

\[ U=\operatorname{span}\{(1,0)\}, \qquad W=\operatorname{span}\{(1,1)\}. \]

Compute the principal angle.

Answer

Normalize $(1,1)$ to $\frac{1}{\sqrt2}(1,1)$. Then

\[ \cos\theta=(1,0)\cdot \frac{1}{\sqrt2}(1,1)=\frac{1}{\sqrt2}. \]

Hence $\theta=\pi/4$.

Task 3: Projection distance

Let principal angles be $\theta_1=0.3$ and $\theta_2=0.8$. Compute the projection distance.

Answer

The projection distance is

\[ d_{\mathrm{projection}}=\sin\theta_2=\sin(0.8)\approx 0.7174. \]

Similar practice question

Let principal angles be $\theta_1=0.1$, $\theta_2=0.4$, and $\theta_3=0.7$. Compute the geodesic and chordal distances.

Answer

\[ d_{\operatorname{Gr}}=\sqrt{0.1^2+0.4^2+0.7^2}\approx 0.8124. \]

\[ d_{\mathrm{chordal}}=\sqrt{\sin^2(0.1)+\sin^2(0.4)+\sin^2(0.7)}\approx 0.7580. \]

# Lab 17: Grassmannians and Distances Between Subspaces {.unnumbered} # Lab goals In this lab you will compute and interpret distances between subspaces. The main computational objects are: - orthonormal basis matrices $Q_U$ and $Q_W$; - the matrix $Q_U^TQ_W$; - singular values $\sigma_i$; - principal angles $\theta_i=\cos^{-1}(\sigma_i)$; - Grassmannian distances such as geodesic and chordal distance. # Python practice notebook The Jupyter notebook version contains guided computations, solutions, and similar practice questions. [Download Lab 17 notebook](lab-17-grassmannians-subspace-distances-independent-study.ipynb) Google Colab link: [Open in Google Colab](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/labs/lab-17-grassmannians-subspace-distances-independent-study.ipynb) # Interactive lab Use the interactive HTML page to visualize principal angles between two planes in $\mathbb R^3$ and to compute Grassmannian distances. [Open Lab 17 Interactive](lab-17-interactive.html) # Background formulas If $Q_U$ and $Q_W$ are orthonormal basis matrices for two $k$-dimensional subspaces, then the principal angles are computed by $$ Q_U^TQ_W=A_0\Sigma B_0^T, \qquad \cos\theta_i=\sigma_i. $$ The Grassmann geodesic distance is $$ d_{\operatorname{Gr}}(U,W)=\left(\sum_{i=1}^k\theta_i^2\right)^{1/2}. $$ The chordal distance is $$ d_{\mathrm{chordal}}(U,W)=\left(\sum_{i=1}^k\sin^2\theta_i\right)^{1/2}. $$ The projection representation is $$ P_U=Q_UQ_U^T. $$ # Independent-study tasks ## Task 1: Two lines Let $$ U=\operatorname{span}\{(1,0)\}, \qquad W=\operatorname{span}\{(1,1)\}. $$ Compute the principal angle. <details> <summary>Answer</summary> Normalize $(1,1)$ to $\frac{1}{\sqrt2}(1,1)$. Then $$ \cos\theta=(1,0)\cdot \frac{1}{\sqrt2}(1,1)=\frac{1}{\sqrt2}. $$ Hence $\theta=\pi/4$. </details> ## Task 2: Two planes sharing a line Let $$ U=\operatorname{span}\{\vec e_1,\vec e_2\}, \qquad W=\operatorname{span}\{\vec e_1,\cos\alpha\,\vec e_2+\sin\alpha\,\vec e_3\}. $$ Find the principal angles and the geodesic distance. <details> <summary>Answer</summary> The subspaces share $\vec e_1$, so $\theta_1=0$. The second direction is rotated by $\alpha$, so $\theta_2=\alpha$. Thus $$ d_{\operatorname{Gr}}(U,W)=\sqrt{0^2+\alpha^2}=\alpha. $$ </details> ## Task 3: Projection distance Let principal angles be $\theta_1=0.3$ and $\theta_2=0.8$. Compute the projection distance. <details> <summary>Answer</summary> The projection distance is $$ d_{\mathrm{projection}}=\sin\theta_2=\sin(0.8)\approx 0.7174. $$ </details> ## Similar practice question Let principal angles be $\theta_1=0.1$, $\theta_2=0.4$, and $\theta_3=0.7$. Compute the geodesic and chordal distances. <details> <summary>Answer</summary> $$ d_{\operatorname{Gr}}=\sqrt{0.1^2+0.4^2+0.7^2}\approx 0.8124. $$ $$ d_{\mathrm{chordal}}=\sqrt{\sin^2(0.1)+\sin^2(0.4)+\sin^2(0.7)}\approx 0.7580. $$ </details>