Math Problems Vault

Problem 5.3.1

(a)(b)(c)(d)

For $x = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}$ , $y = \begin{pmatrix} y_{1} \\ y_{2} \\ y_{3} \end{pmatrix}$ , determine which of the following are inner products for $\mathbb {R}^{3 \times 1}$ .

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(a)

$\left\langle x, y \right\rangle = x_{1}y_{1} + x_{3}y_{3}$ ,

(b)

$\left\langle x, y \right\rangle = x_{1}y_{1} - x_{2}y_{2} + x_{3}y_{3}$ ,

(c)

$\left\langle x, y \right\rangle = 2x_{1}y_{1} + x_{2}y_{2} + 4x_{3}y_{3}$ ,

(d)

$\left\langle x, y \right\rangle = x_{1}^{2}y_{1}^{2} + x_{2}^{2}y_{2}^{2} + x_{3}^{2}y_{3}^{2}$ .

Remark.

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An inner product on a real vector space $V$ is a function $\left\langle x, y \right\rangle$ of ordered pairs of vectors satisfying: $\left\langle x, x \right\rangle$ is real with $\left\langle x, x \right\rangle \geq 0$ , and $\left\langle x, x \right\rangle = 0$ if and only if $x = 0$ ; $\left\langle x, \alpha y \right\rangle = \alpha \left\langle x, y \right\rangle$ for all scalars $\alpha$ ; $\left\langle x, y+z \right\rangle = \left\langle x, y \right\rangle + \left\langle x, z \right\rangle$ ; and $\left\langle x, y \right\rangle = \left\langle y, x \right\rangle$ .

Problem 5.3.2

(a)(b)(c)

For a general inner-product space $V$ , explain why each of the following statements must be true.

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(a)

If $\left\langle x, y \right\rangle = 0$ for all $x \in V$ , then $y = 0$ .

(b)

$\left\langle \alpha x, y \right\rangle = \alpha \left\langle x, y \right\rangle$ for all $x, y \in V$ and for all scalars $\alpha$ .

(c)

$\left\langle x+y, z \right\rangle = \left\langle x, z \right\rangle + \left\langle y, z \right\rangle$ for all $x, y, z \in V$ .

Problem 5.3.3

Let $V$ be an inner-product space with an inner product $\left\langle x, y \right\rangle$ . Explain why the function defined by $\left\| \star \right\| = \sqrt{ \left\langle \star , \star \right\rangle }$ satisfies the first two general-matrix-norm properties $\left\| A\right\| \geq 0$ with $\left\| A\right\| =0 \iff A=0$ , and $\left\| \alpha A\right\| = \left|\alpha \right|\left\| A\right\|$ .

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Problem 5.3.4

For a real inner-product space with $\left\| \star \right\|^{2} = \left\langle \star , \star \right\rangle$ , derive the inequality

\left\langle x, y \right\rangle \leq \frac{\left\| x\right\| ^{2}+\left\| y\right\| ^{2}}{2}.

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Problem 5.3.5

(a)(b)(c)

For $n \times n$ matrices $A$ and $B$ , explain why each of the following inequalities is valid.

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(a)

$\left|\operatorname {trace}(B)\right|^{2} \leq n \, \operatorname {trace}(B^{*}B)$ .

(b)

$\operatorname {trace}(B^{2}) \leq \operatorname {trace}(B^{T}B)$ for real matrices.

(c)

$\operatorname {trace}(A^{T}B) \leq \dfrac {\operatorname {trace}(A^{T}A) + \operatorname {trace}(B^{T}B)}{2}$ for real matrices.

Remark.

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Example 5.3.3: for $A, B \in \mathbb {R}^{m \times n}$ equipped with the standard matrix inner product $\left\langle A, B \right\rangle = \operatorname {trace}(A^{T}B)$ and associated Frobenius norm $\left\| A\right\|_{F} = \sqrt{ \left\langle A, A \right\rangle } = \sqrt{\operatorname {trace}(A^{T}A)}$ , the CBS inequality $\left\langle A, B \right\rangle^{2} \leq \left\| A\right\|_{F}^{2}\left\| B\right\|_{F}^{2}$ gives $\operatorname {trace}(A^{T}B)^{2} \leq \operatorname {trace}(A^{T}A)\operatorname {trace}(B^{T}B)$ .

Problem 5.3.6

Extend the proof of the parallelogram identity's converse to include complex spaces.

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Remark.

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The parallelogram identity states that for a given norm $\left\| \star \right\|$ on a vector space $V$ , there exists an inner product on $V$ such that $\left\langle \star , \star \right\rangle = \left\| \star \right\|^{2}$ if and only if $\left\| x+y\right\|^{2} + \left\| x-y\right\|^{2} = 2\left(\left\| x\right\|^{2}+\left\| y\right\|^{2}\right)$ holds for all $x, y \in V$ . For a real space satisfying this identity, the inner product recovering the norm is $\left\langle x, y \right\rangle = \frac{1}{4}\left(\left\| x+y\right\|^{2} - \left\| x-y\right\|^{2}\right)$ .

Problem 5.3.7

Explain why there does not exist an inner product on $\mathbb {C}^{n}$ ( $n \geq 2$ ) such that $\left\| \star \right\|_{\infty } = \sqrt{ \left\langle \star , \star \right\rangle }$ .

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Problem 5.3.8

Explain why the Frobenius matrix norm on $\mathbb {C}^{n \times n}$ must satisfy the parallelogram identity.

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Problem 5.3.9

For $n \geq 2$ , is either the matrix 1-, 2-, or $\infty$ -norm generated by an inner product on $\mathbb {C}^{n \times n}$ ?

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