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Chapter 8: Q15E (page 413)

If Ais a symmetric matrix such that Av鈫�=3v鈫�and Aw鈫�=4w鈫�, then the equation v鈫�路w鈫�=0must hold.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Step 1: Definition of a symmetric matrix

Let the matrix A is equal to the transpose of the matrix, is said to be symmetric matrix, that is, AT=A.

Also, if a matrix A is a symmetric matrix, then it is orthogonally diagonalizable.

02

Step 2: To Find TRUE or FALSE

It is given that, Av鈫�=3v鈫�andAw鈫�=4w鈫�.

This meansv鈫�andw鈫�are vectors that are inA'sorthogonal basis.

Sincev鈫�andw鈫�are part of an orthogonal basis so they must be perpendicular and therefore,v鈫�路w鈫�=0.

Thus, the given statement is TRUE.

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Most popular questions from this chapter

For ndistinct scalars a1,a2,...,an, find

role="math" localid="1659609991385" det[a1a2鈥�ana12a22鈥�an2鈰�鈰�鈰�a1na2n鈥�ann]

Let 位 be a real eigenvalue of an n x n matrix A. Show that

蟽n猢�|位|猢�蟽1,

where蟽1and蟽nare the largest and the smallest singular values of A, respectively.

True or false? If Ais a symmetric matrix, thenrank(A)=rank(A2)

Find the dimension of the spaceQn of all quadratic forms in n variables.

Diagonalize the n脳nmatrix

[100鈰�01010鈰�10001鈰�00鈰�鈰�鈰�鈰�鈰�鈰�010鈰�10100鈰�01].

(All ones along both diagonals, and zeros elsewhere.)
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