Q37E Consider a symmetric nxn聽matrix... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q37E (page 346)

Consider a symmetric nxnmatrix A.
a. Show that if $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two vectors in ${鈩�}^{n}$ , then
$A \overset{鈫�}{v} . \overset{鈫�}{w} = \overset{鈫�}{v} . A \overset{鈫�}{w} .$
b. Show that if $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two eigenvectors of A, with distinct eigenvalues, then $\overset{鈫�}{w}$ is orthogonal to $\overset{鈫�}{v}$ .

Short Answer

Expert verified

The part(a), part(b) is

$(a) A \overset{鈫�}{v} . \overset{鈫�}{w} = \overset{鈫�}{v} . A \overset{鈫�}{w} . two results are the same . (b) Since 位_{1} 鈮� 位_{2} and v . w = 0$

Step by step solution

Solving the matrixPart (a)

The i-th component of is $Av is {鈭�}_{j = 1}^{n} a_{i, j}$ i, and the i-th component of is $w_{i}$ .So,

$A \overset{鈫�}{v} . \overset{鈫�}{w} = {鈭�}_{i = 1}^{n} ({鈭�}_{i = 1}^{n} a_{ij} v_{j}) w_{i} {鈭�}_{i = 1}^{n} w_{i} {鈭�}_{j = 1}^{n} a_{ij} v_{j}$

Similarly, we compute

role="math" localid="1659594596446" $\overset{鈫�}{v} . A \overset{鈫�}{w} = {鈭�}_{i = 1}^{n} v_{i} {鈭�}_{j = 1}^{n} a_{ij} v_{j}$

That these two results are the same, we conclude from distributivity.

Part (b)

Let $Av = 位_{1} v$ and $Aw = 位_{2} w,$ ,

where $位_{1} 鈮� 位_{2} .$ .

Now, using part a., we have $Av . w = v . aw 鈬� 位_{1} v . w = v . 位_{2} w 鈬� 位_{1} (v . w) = 位_{2} (v . w) . since 位_{1} 鈮� 位_{2}, then v . w = 0$ .

Therefore, the final result of part(a), part(b) is

$(a) A \overset{鈫�}{v} . \overset{鈫�}{w} = \overset{鈫�}{v} . A \overset{鈫�}{w} . two results are the same . (b) Since 位_{1} 鈮� 位_{2} and v . w = 0$

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Short Answer

Step by step solution

Solving the matrixPart (a)

Part (b)

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91影视

Consider a symmetric nxnmatrix A.a. Show that ifv鈫�andw鈫�are two vectors in鈩�n, then Av鈫�.w鈫�=v鈫�.Aw鈫�. b. Show that ifv鈫�andw鈫�are two eigenvectors of A, with distinct eigenvalues, thenw鈫�is orthogonal tov鈫�.

Short Answer

Step by step solution

Solving the matrixPart (a)

Part (b)

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.