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a. Suppose A is a symmetric n n matrix. Using the Spectral Theorem, prove that if Ax x =
Chapter 6, Problem 9(choose chapter or problem)
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.
Questions & Answers
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