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

If your book is hosted on GitHub, you can also add a 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) If your book is hosted on GitHub, you can also add a 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>