Math Problems Vault

Problem 5.15.1

(a)(b)

Determine the angles $\theta_{\min }$ and $\theta_{\max }$ between the following subspaces of $\mathbb {R}^{3}$ .

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(a)

$M = xy\text{-plane}$ , $N = \left\langle (1,0,0),(0,1,1)\right\rangle$ .

(b)

$M = xy\text{-plane}$ , $N = \left\langle (0,1,1)\right\rangle$ .

Problem 5.15.2

(a)(b)

Determine the principal angles between the following subspaces of $\mathbb {R}^{3}$ .

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(a)

$M = xy\text{-plane}$ , $N = \left\langle (1,0,0),(0,1,1)\right\rangle$ .

(b)

$M = xy\text{-plane}$ , $N = \left\langle (0,1,1)\right\rangle$ .

Remark.

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The principal angles between $M$ and $N$ are defined recursively: $\theta_{1} = \theta_{\min }$ , with $u_{1}, v_{1}$ the unit vectors attaining the maximum in $\cos \theta_{\min } = v_{1}^{T}u_{1}$ ; then $M_{2} = u_{1}^{\perp } \cap M$ , $N_{2} = v_{1}^{\perp } \cap N$ , and $\theta_{2}$ is the minimal angle between $M_{2}$ and $N_{2}$ , and so on.

Problem 5.15.3

(a)(b)(c)

Let $\theta_{\min }$ be the minimal angle between nonzero subspaces $M, N \subseteq \mathbb {R}^{n}$ .

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(a)

Explain why $\theta_{\max }=0$ if and only if $M=N$ .

(b)

Explain why $\theta_{\min }=0$ if and only if $M\cap N \neq 0$ .

(c)

Explain why $\theta_{\min }=\pi /2$ if and only if $M\perp N$ .

Remark.

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Equation (5.15.16): the maximal angle satisfies $\sin \theta_{\max } = \operatorname {gap}(M,N) = \left\| P_{M}-P_{N}\right\|_{2}$ .

Problem 5.15.4

Let $\theta_{\min }$ be the minimal angle between nonzero subspaces $M, N \subset \mathbb {R}^{n}$ , and let $\theta_{\min }^{\perp }$ denote the minimal angle between $M^{\perp }$ and $N^{\perp }$ . Prove that if $M\oplus N = \mathbb {R}^{n}$ , then $\theta_{\min } = \theta_{\min }^{\perp }$ .

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Remark.

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Equation (5.15.4): if $M, N$ are complementary, $\sin \theta_{\min } = 1/\left\| (P_{M} - P_{N})^{-1}\right\|_{2}$ .

Problem 5.15.5

For nonzero subspaces $M, N \subset \mathbb {R}^{n}$ , let $\tilde{\theta }_{\min }$ denote the minimal angle between $M$ and $N^{\perp }$ , and let $\theta_{\max }$ be the maximal angle between $M$ and $N$ . Prove that if $M \oplus N^{\perp } = \mathbb {R}^{n}$ , then $\cos \tilde{\theta }_{\min } = \sin \theta_{\max }$ .

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Remark.

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Problem 5.15.6

For subspaces $M, N \subseteq \mathbb {R}^{n}$ , prove that $P_{M}-P_{N}$ is nonsingular if and only if $M$ and $N$ are complementary.

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Remark.

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Equation (5.15.7): with U = [U1 | U2], V = [V1 | V2] orthogonal matrices whose columns give orthonormal bases for M, M-perp, N-perp, N respectively (so that P_M = U1 U1^T, P_N = V2 V2^T), U^T (P_M - P_N) V = block-diag(U1^T V1, -U2^T V2).

Problem 5.15.7

(a)(b)(c)

For complementary spaces $M, N \subset \mathbb {R}^{n}$ , let $P = P_{MN}$ be the oblique projector onto $M$ along $N$ , and let $Q = P_{M^{\perp }N^{\perp }}$ be the oblique projector onto $M^{\perp }$ along $N^{\perp }$ .

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(a)

Prove that $(P_{M}-P_{N})^{-1} = P-Q$ .

(b)

If $\theta_{\min }$ is the minimal angle between $M$ and $N$ , explain why $\sin \theta_{\min } = 1/\left\| P-Q\right\|_{2}$ .

(c)

Explain why $\left\| P-Q\right\|_{2} = \left\| P\right\|_{2}$ .

Problem 5.15.8

Prove that if $f: V \to \mathbb {R}$ is a function defined on a space $V$ such that $f(\alpha x) = \alpha f(x)$ for scalars $\alpha \geq 0$ , then $\max_{\left\| x\right\| =1} f(x) = \max_{\left\| x\right\| \leq 1} f(x)$ .

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Problem 5.15.9

(a)(b)

Let $M$ and $N$ be nonzero complementary subspaces of $\mathbb {R}^{n}$ .

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(a)

Explain why $P_{MN} = \left[(I-P_{N})P_{M}\right]^{\dagger }$ , where $P_{M}$ and $P_{N}$ are the orthogonal projectors onto $M$ and $N$ , respectively, and $P_{MN}$ is the oblique projector onto $M$ along $N$ .

(b)

If $\theta_{\min }$ is the minimal angle between $M$ and $N$ , explain why

\sin \theta _{\min } = \left\| \left[(I-P_{N})P_{M}\right]^{\dagger }\right\| _{2}^{-1} = \left\| \left[P_{M}(I-P_{N})\right]^{\dagger }\right\| _{2}^{-1} = \left\| \left[(I-P_{M})P_{N}\right]^{\dagger }\right\| _{2}^{-1} = \left\| \left[P_{N}(I-P_{M})\right]^{\dagger }\right\| _{2}^{-1}.

Problem 5.15.10

(a)(b)

For complementary subspaces $M, N \subset \mathbb {R}^{n}$ , let $\theta_{\min }$ be the minimal angle between $M$ and $N$ , and let $\bar{\theta }_{\min }$ denote the minimal angle between $M$ and $N^{\perp }$ .

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(a)

If $P_{MN}$ is the oblique projector onto $M$ along $N$ , prove that $\cos \bar{\theta }_{\min } = \left\| P_{MN}^{\dagger }\right\|_{2}$ .

(b)

Explain why $\sin \theta_{\min } \leq \cos \bar{\theta }_{\min }$ .

Problem 5.15.11

(a)(b)

Let $U = \left[U_{1}\mid U_{2}\right]$ and $V = \left[V_{1}\mid V_{2}\right]$ be the orthogonal matrices whose columns give orthonormal bases for $M, M^{\perp }, N^{\perp }, N$ respectively (as in the proof of (5.15.3)-(5.15.4)).

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(a)

Prove that if $U_{2}^{T}V_{2}$ is nonsingular, then $\dfrac {1}{\left\| (U_{2}^{T}V_{2})^{-1}\right\|_{2}^{2}} = 1 - \left\| U_{2}^{T}V_{1}\right\|_{2}^{2}$ .

(b)

Prove that if $U_{2}^{T}V_{1}$ is nonsingular, then $\left\| U_{2}^{T}V_{2}\right\|_{2}^{2} = 1 - \dfrac {1}{\left\| (U_{2}^{T}V_{1})^{-1}\right\|_{2}^{2}}$ .