Due on Friday, March 8.
Prove that unitary equivalence is an equivalence relation.
Prove that if a real 2-by-2 matrix has a complex number \(\lambda\) as an eigenvalue, then it must also have \(\bar{\lambda}\) as an eigenvalue. Hint: What can you say about the characteristic polynomial of an arbitrary real 2-by-2 matrix?
Prove that if \(U\) is a real 2-by-2 unitary matrix with \(\det U = -1\), then \(U\) has eigenvalues \(1\) and \(-1\).
Prove that if \(U\) is a real 2-by-2 unitary matrix with \(\det U = -1\), then the eigenspaces corresponding to \(+1\) and \(-1\) are orthogonal.
Prove that if \(U\) is a real 2-by-2 unitary matrix with \(\det U = -1\), then \(U\) is unitary equivalent to the diagonal matrix with diagonal entries \(+1\) and \(-1\).