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Chapter 3: Q27E (page 110)

If \(\overrightarrow v \) and \(\overrightarrow w \) are linearly independent eigenvectors of a symmetric matrix \(A\), then \(\overrightarrow w \) must be orthogonal to \(\overrightarrow v \).

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Given information:

The given statement is: If\(\overrightarrow v \)and\(\overrightarrow w \)are linearly independent eigenvectors of a symmetric matrix\(A\), then\(\overrightarrow w \)must be orthogonal to\(\overrightarrow v \).

02

Check whether the given statement is TRUE or FALSE

Spectral theorem: A matrix\(A\)is orthogonally diagonalizable if and only if\(A\)is symmetric.

By the above theorem, if \(\overrightarrow v \) and \(\overrightarrow w \) are linearly independent columns of the symmetric matrix \(A\), then they are orthogonal.

03

Final Answer

The given statement is TRUE.

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|a11x1+a12x2+鈥�+a1mxm=0a21x1+a22x2+鈥�+a2mxm=0鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆��鈰�鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆��鈰�鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌�鈰�鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌�鈰�鈥�an1x1+a22x2+鈥�+anmxm=0|.

What is the relationship between the dimension of Vand the quantitym-n

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