Show that if is the only eigenvalue of a symmetric matrix A, then A = I

Chapter 6, Problem 7

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QUESTION:

Show that if is the only eigenvalue of a symmetric matrix A, then A = I.

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QUESTION:

Show that if is the only eigenvalue of a symmetric matrix A, then A = I.

ANSWER:

Step 1 of 2

Given that is an only eigenvalue of a symmetric matrix A.

To prove that

Since A is symmetric, by the spectral theorem, all the eigenvalues of A are real and there is an orthogonal matrix such that is diagonal.

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