a. Suppose A is a symmetric n n matrix. Using the Spectral Theorem, prove that if Ax x =

Chapter 6, Problem 9

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QUESTION:

a. Suppose A is a symmetric n n matrix. Using the Spectral Theorem, prove that if Ax x = 0 for every vector x Rn, then A = O. b. Give an example to show that the hypothesis of symmetry is needed in part a.

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QUESTION:

a. Suppose A is a symmetric n n matrix. Using the Spectral Theorem, prove that if Ax x = 0 for every vector x Rn, then A = O. b. Give an example to show that the hypothesis of symmetry is needed in part a.

ANSWER:

Step 1 of 3

(a)By the use of Spectral theorem, Let  be a symmetric  matrix. Then,

1.The eigenvalues of  are real.

2.There is an orthonormal basis  for  consisting of eigenvectors of .

That is, there is an orthogonal matrix  so that  is diagonal

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