in this problem, answer \"True\" or \"False\" for each question. Note: there is

Question

in this problem, answer "True" or "False" for each question.

Note: there is no partial credit for this problem. You must answer all parts correctly to receive credit. You will not be shown the correct answers for individual parts.

1. If A is symmetric and P?1APis diagonal, then PP is orthogonal.
True
False

2. If A is a symmetric matrix, then any two eigenvectors of A are orthogonal to each other.
True
False

3. For a vector v in a complex inner product space, ?v,v> is always a real number.
True
False

4. If v,w are vectors in a complex inner product space, then ?2v,w?=2?v,w?
True
False

TTTT and TTTF are both incorrect.

Explanation / Answer

1. If A is symmetric and P-1 AP is diagonal, then P is orthogonal.

True

If A is an symmetric matrix, then the eigenvectors of A associated with distinct eigenvalues are orthogonal.Further, since A is diagonalizable, all the the eigenvectors of A are distinct and linearly independent.Also, if A is real symmetric then A has an orthonormal basis of real eigenvectors.

2. If A is a symmetric matrix, then any two eigenvectors of A are orthogonal to each other.

True

Please see 1. above.

3. For a vector v in a complex inner product space, <v,v> is always a real number.

True

If v = a+ib, then <v,v> = a2+(ib)2 = a2-b2 which is always a real number.

4. If v,w are vectors in a complex inner product space, then ?2v,w?=2?v,w?

True

As per the Homogeneity axiom , for any scalar k, <ku, v> = k<u, v>]

TTTT

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