Q42E Consider a complex n 脳 m matrix... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q42E (page 374)

Consider a complex n 脳 m matrix A. Theconjugate $\overset{炉}{A}$ isdefined by taking the conjugate of each entry of A. For example, if
$A = [\begin{matrix} 2 + 3 i & 5 \\ 2 i & 9 \end{matrix}], 鈥� t h e n \overset{炉}{A} = [\begin{matrix} 2 - 3 i & 5 \\ - 2 i & 9 \end{matrix}]$
a. Show that if A and B are complex n 脳 p and p 脳 m matrices, respectively, then $\overset{炉}{A B} = \overset{炉}{A} \overset{炉}{B}$
b. Let A be a real n 脳 n matrixand $\overset{鈫�}{v} + i \overset{鈫�}{w}$ aneigenvector of A with eigenvalue p + iq. Show that the vector $\overset{鈫�}{v} - i \overset{鈫�}{w}$ is an eigenvector of A with eigenvalue p 鈭� iq.

Short Answer

Expert verified

a. (i,j)-th entry of = $\overset{炉}{A B} = \overset{炉}{A} \overset{炉}{B}$ = ${鈭�}_{k = 1}^{p} \overset{炉}{a_{i, k}} 鈰� \overset{炉}{b_{k, j}} = {鈭�}_{k = 1}^{p} \overset{炉}{a_{i, k} b_{k, j}} = \overset{炉}{{鈭�}_{k = 1}^{p} a_{i, k} b_{k, j}}$

b. $v 鈭� i w$ is also an eigenvector of A, with the eigen value p-iq.

Step by step solution

Step 1:The (i , j)-th entry ofAB¯

$\overset{炉}{{鈭�}_{k = 1}^{p} a_{i, k} b_{k, j}}$

On the other hand, the (i ,j)-the entry of is

${鈭�}_{k = 1}^{p} \overset{炉}{a_{i, k}} 鈰� \overset{炉}{b_{k, j}} = {鈭�}_{k = 1}^{p} \overset{炉}{a_{i, k} b_{k, j}} = \overset{炉}{{鈭�}_{k = 1}^{p} a_{i, k} b_{k, j}}$

b.Let A be a real n 脳 n matrixand an eigenvector of A with eigenvalue p + iq. Show that the vectoris an eigenvector of A with eigenvalue p 鈭� iq.

Finding eigenvector of A

Since A is a real matrix, then. $\overset{炉}{A} = A$

If $v + i u$ is an eigen vector of A with the eigen value,p+iq, then we have

$A (v 鈭� i w) = A \overset{炉}{v + i w} = \overset{炉}{A (v + i w)} = \overset{炉}{(p + i q) (v + i w)} = \overset{炉}{p + i q} 鈰� \overset{炉}{v + i w} = (p 鈭� i q) (v 鈭� i w)$

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Short Answer

Step by step solution

Step 1:The (i , j)-th entry ofAB¯

Finding eigenvector of A

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