restgarnut
2022-09-12
Answered

Why ${a}^{H}\text{diag}(b{b}^{H})a={b}^{H}\text{diag}(a{a}^{H})b,a,b\in {\mathbb{C}}^{N\times 1}$?

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Salvador Howard

Answered 2022-09-13
Author has **12** answers

$\text{diag}(A)=\sum _{i=1}^{N}{e}_{i}{e}_{i}^{H}A{e}_{i}{e}_{i}^{H}$ , where ei is the standard basis of ${\mathbb{C}}^{N}$

Therefore

$$\begin{array}{rl}{a}^{H}\text{diag}\left(b{b}^{H}\right)a& =\sum _{i=1}^{N}{a}^{H}{e}_{i}{e}_{i}^{H}b{b}^{H}{e}_{i}{e}_{i}^{H}a\\ & =\sum _{i=1}^{N}{b}^{H}{e}_{i}{e}_{i}^{H}a{a}^{H}{e}_{i}{e}_{i}^{H}b\\ & ={b}^{H}\text{diag}\left(a{a}^{H}\right)b.\end{array}$$

Therefore

$$\begin{array}{rl}{a}^{H}\text{diag}\left(b{b}^{H}\right)a& =\sum _{i=1}^{N}{a}^{H}{e}_{i}{e}_{i}^{H}b{b}^{H}{e}_{i}{e}_{i}^{H}a\\ & =\sum _{i=1}^{N}{b}^{H}{e}_{i}{e}_{i}^{H}a{a}^{H}{e}_{i}{e}_{i}^{H}b\\ & ={b}^{H}\text{diag}\left(a{a}^{H}\right)b.\end{array}$$

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Let B be a 4x4 matrix to which we apply the following operations:

1. double column 1,

2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

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2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

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The

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Write out the system of equations that corresponds
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(b)$\left(\begin{array}{ccccc}5& -2& 1& |& 3\\ 2& 3& -4& |& 0\end{array}\right)$

(c)$\left(\begin{array}{ccccc}2& 1& 4& |& -1\\ 4& -2& 3& |& 4\\ 5& 2& 6& |& -1\end{array}\right)$

(d)$\left(\begin{array}{cccccc}4& -3& 1& 2& |& 4\\ 3& 1& -5& 6& |& 5\\ 1& 1& 2& 4& |& 8\\ 5& 1& 3& -2& |& 7\end{array}\right)$

(a)

(b)

(c)

(d)

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a)

b)

c)

(a),(b),(c) need to be solved