Q9P Write equations (10.6) to (10.9)... [FREE SOLUTION]

Chapter 3: Q9P (page 148)

Write equations (10.6) to (10.9) in matrix form as discussed just after (10.9).

Short Answer

Expert verified

Inner product of $A$ and $B = {鈭�}_{i = 1}^{n} {A_{i}}^{*} B_{i}$

Norm of $A = A = \sqrt{{鈭�}_{i = 1}^{n} A_{i}^{*} A_{i}}$

$A$ and $B$ are orthogonal $= {鈭�}_{i = 1}^{n} {A_{i}}^{*} B_{i} = 0$

Schwarz inequality $= |{鈭�}_{i = 1}^{n} A_{i}^{*} B_{i}| 猢� \sqrt{{鈭�}_{i = 1}^{n} A_{i}^{*} A_{i}} \sqrt{{鈭�}_{i = 1}^{n} B_{i}^{*} B_{i}}$

Inner product of $A$ and $B =$ $(\begin{matrix} a_{11} & a_{12} & . & a_{1 n} \\ a_{21} & a_{22} & . & a_{2 n} \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & a_{n n} \end{matrix}) . (\begin{matrix} b_{11} & b_{12} & . & b_{1 n} \\ b_{21} & b_{22} & . & b_{2 n} \\ . & . & . & . \\ b_{n 1} & b_{n 2} & . & b_{n n} \end{matrix})$

Norm of A= $d e t |(\begin{matrix} a_{11} & a_{12} & . & a_{1 n} \\ a_{21} & a_{22} & . & a_{2 n} \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & a_{n n} \end{matrix})| = a_{11} |\begin{matrix} a_{22} & . & a_{2 n} \\ . & . & . \\ a_{2 n} & . & a_{n n} \end{matrix}| + a_{12} |\begin{matrix} a_{21} & . & a_{2 n} \\ . & . & . \\ a_{n 1} & . & a_{n n} \end{matrix}| + . . . . . . + a_{1 n} |\begin{matrix} a_{21} & a_{22} & . \\ . & . & . \\ a_{n 1} & a_{n 2} & . \end{matrix}|$

A and B are orthogonal, $(\begin{matrix} a_{11} & a_{12} & . & a_{1 n} \\ a_{21} & a_{22} & . & a_{2 n} \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & a_{n n} \end{matrix}) . (\begin{matrix} b_{11} & b_{12} & . & b_{1 n} \\ b_{21} & b_{22} & . & b_{2 n} \\ . & . & . & . \\ b_{n 1} & b_{n 2} & . & b_{n n} \end{matrix}) = 0$

Schwarz inequality $= |{鈭�}_{i = 1}^{n} A_{i}^{*} B_{i}| 猢� \sqrt{{鈭�}_{i = 1}^{n} A_{i}^{*} A_{i}} \sqrt{{鈭�}_{i = 1}^{n} B_{i}^{*} B_{i}}$

Step by step solution

Given information from question

Inner product:

An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.

More precisely, for a real vector space, an inner product $鉄� 路, 路鉄�$ satisfies the following four properties. Let $u, v$ and $w$ be vectors and $a$ be a scalar, then:

1. $鉄� u + v, w 鉄� = 鉄� u, w 鉄� + 鉄� v, w 鉄�$

2. $鉄� 伪 v, w 鉄� = 伪鉄� v, w 鉄�$

3. $鉄� v, w 鉄� = 鉄� w, v 鉄�$

4. $鉄� v, v 鉄� 猢� 0$ and equal if and only if $v = 0$

Norm: A norm is a function from a real or complex vector space to the non-negative real numbers that exhibits some characteristics similar to the angle to the origin, including scaling, a form of the triangle inequality, and being zero exclusively at the origin.

Norm of $A = A = \sqrt{{鈭�}_{i = 1}^{n} A_{i}^{*} A_{i}}$

$A$ and $B$ are orthogonal $= {鈭�}_{i = 1}^{n} {A_{i}}^{*} B_{i} = 0$

Schwarz inequality $= |{鈭�}_{i = 1}^{n} A_{i}^{*} B_{i}| 猢� \sqrt{{鈭�}_{i = 1}^{n} A_{i}^{*} A_{i}} \sqrt{{鈭�}_{i = 1}^{n} B_{i}^{*} B_{i}}$

Norm of A = $d e t |(\begin{matrix} a_{11} & a_{12} & . & a_{1 n} \\ a_{21} & a_{22} & . & a_{2 n} \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & a_{n n} \end{matrix})| = a_{11} |\begin{matrix} a_{22} & . & a_{2 n} \\ . & . & . \\ a_{n 2} & . & a_{n n} \end{matrix}| + a_{12} |\begin{matrix} a_{21} & . & a_{2 n} \\ . & . & . \\ a_{n 1} & . & a_{n n} \end{matrix}| + . . . . . . . + a_{1 n} |\begin{matrix} a_{21} & a_{22} & . \\ . & . & . \\ a_{n 1} & a_{n 2} & . \end{matrix}|$

A and B are orthogonal , $(\begin{matrix} a_{11} & a_{12} & . & a_{1 n} \\ a_{21} & a_{22} & . & a_{2 n} \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & a_{n n} \end{matrix}) . (\begin{matrix} b_{11} & b_{12} & . & b_{1 n} \\ b_{21} & b_{22} & . & b_{2 n} \\ . & . & . & . \\ b_{n 1} & b_{n 2} & . & b_{n n} \end{matrix}) = 0$

Schwarz inequality $= |{鈭�}_{i = 1}^{n} A_{i}^{*} B_{i}| 猢� \sqrt{{鈭�}_{i = 1}^{n} A_{i}^{*} A_{i}} \sqrt{{鈭�}_{i = 1}^{n} B_{i}^{*} B_{i}}$

$d e t |(\begin{matrix} a_{11} & a_{12} & . & a_{1 n} \\ a_{21} & a_{22} & . & a_{2 n} \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & a_{n n} \end{matrix})| . |(\begin{matrix} b_{11} & b_{12} & . & b_{1 n} \\ b_{21} & b_{22} & . & b_{2 n} \\ . & . & . & . \\ b_{n 1} & b_{n 2} & . & b_{n n} \end{matrix})| 鈮�$

$\sqrt{d e t |(\begin{matrix} a_{11} & a_{12} & . & a_{1 n} \\ a_{21} & a_{22} & . & a_{2 n} \\ . & . & . & . \\ a_{n 1} & a_{n 2} & . & a_{n n} \end{matrix})|} . \sqrt{d e t |(\begin{matrix} b_{11} & b_{12} & . & b_{1 n} \\ b_{21} & b_{22} & . & b_{2 n} \\ . & . & . & . \\ b_{n 1} & b_{n 2} & . & b_{n n} \end{matrix})|}$

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Write equations (10.6) to (10.9) in matrix form as discussed just after (10.9).

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