Hello I need help in this matrix problem. I know it seems pretty simple but I am having hard time proving it..
Let U E M(nxn) (C) be a unitary matrix, that is UU*=U*U=I. Let A, X be eigenpair for U. Prove that |A| = 1
---This is what I have got. So please tell me if I need to add something or is it completely wrong approach. Thanks for the kind help!
Let U be the unitary matrix...
then, Suppose x is an eigenvector corresponding to the eigenvalue λ of U. Then Ux = λx, so
||Ux|| = ||λx|| = |λ| . ||x||
But U preserves lengths, so ||Ux|| = ||x||, and hence |λ| = 1....Do I have to give the reason why U preserves lengths?? and how??
Only you know if you need to include in the proof that orthogonal matrices preserve lengths. If you do need to prove that, then consider
and so on.