I was given the following question, but I can't get to prove it, and I would like to have some kind of hint to the answer. The question is:
Let A,B be two matrices 2X2 with real numbers, NOT diagonals. If (A^t)A = I = (B^t)B and A,B are diagonalizable so A and B are similar.
What I was trying to think about: I know that (A^t)A = I, so I can get two important pieces of information:
- A,B are unitary. So the abs of thier eigenvalues is 1.
- 1 is an eigenvalue for (A^t)A and for (B^t)B.
But I can't connect the pieces of information to give me the reason to believe that A is similar to B. I can guess that 1 is an eigenvalue for A and B, but I still have one eigenvalue to find for both of them to prove that the sentence is right.
Can anyone please hint me to the answer? Big thanks in advance.