4. Let 1 0 -1 2 3 0130 A=1-1 3 0 2 2 2 2 2 5 -2 3 0 2 -2 6 (a) [10 marks] Define

Question

4. Let 1 0 -1 2 3 0130 A=1-1 3 0 2 2 2 2 2 5 -2 3 0 2 -2 6 (a) [10 marks] Define and compute the1 operator norm of A. (b) [7 marks] Prove that any eigenvalues of A are real. (c) [8 marks] Prove that any eigenvectors of A corresponding to different eigenvalues are orthogonal to each other (in the standard, Euclidean inner product) Hint: (b) and (c) can be answered without computing explicitly the eigenvalues /eigenvec tors of A. In fact, you only need to use one property of A, which can be easily checked. Total: 25 marks

Explanation / Answer

It may be observed that A T = A so that A is a real symmetric matrix.

b). The eigenvalues of real symmetric matrices are real. Hence, A has real eigenvalues.

c). The eigenvectors of real symmetric matrices corresponding to different eigenvalues are mutually orthogonal. Hence, the eigenvectors of A corresponding to different eigenvalues are orthogonal to one another.

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