
orthogonal matrix...
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
http://quicklatex.com/cache3/ql_1dc4...ce4d5e8_l3.png
and so on.