Q50E TRUE OR FALSEIf a 3脳3matrix A r... [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 5: Q50E (page 264)

TRUE OR FALSE
If a $3 脳 3$ matrix A represents the orthogonal projection onto a plane V in $R^{3}$ , then there must exists an orthogonal $3 脳 3$ matrix S such that $S^{T} A S$ is diagonal.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Explanation of the solution

Consider an orthogonal basis for the subspace V, say $\{{\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}\}$ and ${\overset{鈫�}{v}}_{3}$ is a vector perpendicular to the plane V.

Since A represents the orthogonal projection onto the plane V as follows:

role="math" localid="1659629274695" $A [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� {\overset{鈫�}{v}}_{3}] = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� 0] A S = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� 0] 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� w h e r e 鈥� 鈥� S = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� {\overset{鈫�}{v}}_{3}] A S = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� {\overset{鈫�}{v}}_{3}] [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] A S = S [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Simplify further as follows.

$A S = S [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] S^{- 1} A S = S^{- 1} S [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Since S is orthogonal is $S^{T} = S^{- 1}$ .

$S^{T} A S = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

This implies that $S^{T} A S$ is a diagonal matrix.

Thus, the given statement there exits an orthogonal $3 脳 3$ matrix such that $S^{T} A S$ is diagonal鈥� is true.

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91影视

TRUE OR FALSE
If a $3 脳 3$ matrix A represents the orthogonal projection onto a plane V in $R^{3}$ , then there must exists an orthogonal $3 脳 3$ matrix S such that $S^{T} A S$ is diagonal.

Short Answer

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91影视

TRUE OR FALSEIf a 3脳3matrix A represents the orthogonal projection onto a plane V in R3, then there must exists an orthogonal 3脳3matrix S such that STASis diagonal.

Short Answer

Step by step solution

Explanation of the solution

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Probability and Statistics

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Statistics

Geometry

Logic and Functions

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TRUE OR FALSE
If a $3 脳 3$ matrix A represents the orthogonal projection onto a plane V in $R^{3}$ , then there must exists an orthogonal $3 脳 3$ matrix S such that $S^{T} A S$ is diagonal.