Indicate whether each statement is always true or some times false. Justify your
ID: 2940336 • Letter: I
Question
Indicate whether each statement is always true or some times false. Justify your answer by giving a logical argument or a counterexample. If A is a square matrix, then AAT and ATA are orthogonally diagonalizable. If v1 and v2 are eigenvectors from distinct eigenspaces of a symmetric matrix, then ||v1+v2||2 = ||v1||2 + ||v1||2. An orthogonal matrix is orthogonally diagonalizable. If A is an invertible orthogonally diagonalizable matrix, then A-1 is orthogonally diagonalizable.Explanation / Answer
(a) Yes. This is true for any real symmetric matrix, and for anymatrix A, AAT and ATA are both symmetric. (b) Yes. Suppose , are distinct eigenvalues foreigenvectors v,u respectively. Then (Au,v)=(u,v)=(u,v). Also, (Au,v)=(u,ATv)=(u,Av) (since A is symmetric) =(u,v)=(u,v). Thus, (u,v)=(u,v) and since , are distinct,it follows that (u,v)=0. Hence, || u+v ||2 = || u||2 + || v ||2 + 2(u,v) = || u ||2 + || v ||2 . (c) No. A matrix is orthogonally diagonalizable => A issymmetric. Now if A is orthogonal and symmetric, then AAT = A2 = I. But rotation by any non-trivialangle in R2 gives an orthogonal matrixwhich is not necessarily square. (d) Yes. If PAPT=D and all the diagonal entries of D arenon-zero, then taking inverses on both sides, we get (PT)-1A-1P-1 =D-1. But note that (PT)-1=P sinceP is orthogonal and P-1=PT.