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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: Q39E (page 264)

TRUE OR FALSE
Determine whether the statement 鈥淭he matrix A is symmetric and matrix S is orthogonal, then matrix $S^{- 1} A S$ must be symmetric.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of the symmetric matrix

鈥淢atrix A is symmetric if $A^{T} = A$ . Matrix B is orthogonal if and only if $B^{T} = B^{- 1}$ 鈥�

02

Explanation of the solution

Consider the statement as follows:

鈥淭he matrix A is symmetric and matrix S is orthogonal then the matrix $S^{- 1} A S$ is symmetric.

Consider the matrix as follows:

$Q = S^{- 1} A S$

Taking transpose on both sides as follows.

$Q = S^{- 1} A S Q^{T} = {(S^{- 1} A S)}^{T} Q^{T} = S^{T} A^{T} {(S^{- 1})}^{T} 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� (1)$

Since, A is symmetric matrix, then

$A^{T} = A$ ......(2)

SinceS is orthogonal as follows:

$S^{T} = S^{- 1} {(S^{- 1})}^{T} = S 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� (3)$

Using (2) and (3) in (1) as follows:

$Q^{T} = S^{- 1} A S Q^{T} = Q Q = S^{- 1} A S$

Hence, $Q = S^{- 1} A S$ is symmetric.

Thus, the given statement 鈥淢atrix A is symmetric if $A^{T} = A$ . Matrix B is orthogonal is and only if $B^{T} = B^{- 1}$ " is true.

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Most popular questions from this chapter

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -B.

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