Question

3. Let T (V), and B be an orthonormal basis, so that M(T,B) (5+20 pts) Is T self-adjoint? Why/Why Not? (5+20 pts) Is T normal? Why/Why Not? . (10 pts/box with explanation) Now, let R E L(V) be a self-adjoint operator, SEL(V) a normal operator, and U E L(V) an operator that is neither self-adjoint nor normal; what properties do these operators have-mark R (if true only for F = R) / C (if true only for F = C) / F (always true) no sometimes) op=R, S, or U R, Self Adjoint S, Normal (Not Self-Adjoint)1 I U, Neither ョortho-normal basis M(op, B) is upper triangular ョortho-normal basis M(op, B) is diagonal all eigenvalues of op are real op has positive square root op has polar decomposition op has singular value decomposition On the next page (not in the boxes!!!)for each answer in the grid give a SHORT EXPLANA- TION (why/why not, or what is the "extra condition(s)" needed for the "sometimes" properties?

Answer 1

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