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Math / Linear Algebra: A Geometric Approach 2 / Chapter 6.4 / Problem 15

Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams

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Suppose A and B are symmetric and AB = BA. Prove there is an orthogonal matrix Q so that

Chapter 6, Problem 15

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QUESTION:

Suppose A and B are symmetric and AB = BA. Prove there is an orthogonal matrix Q so that both Q 1AQ and Q 1BQ are diagonal. (Hint: Let be an eigenvalue of A. Use the Spectral Theorem to show that there is an orthonormal basis for E() consisting of eigenvectors of B.)

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QUESTION:

Suppose A and B are symmetric and AB = BA. Prove there is an orthogonal matrix Q so that both Q 1AQ and Q 1BQ are diagonal. (Hint: Let be an eigenvalue of A. Use the Spectral Theorem to show that there is an orthonormal basis for E() consisting of eigenvectors of B.)

ANSWER:

Step 1 of 2

Given that and are symmetric matrices with .

To prove there is an orthogonal matrix such that both and are diagonal.

Since is symmetric, by spectral theorem, eigenvalues of are real and there is an orthonormal basis of consisting of eigenvectors of .

Now, suppose be an eigenvector of corresponding to an arbitrary eigenvalue . Then, by definition,

.

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