(Friedberg 6.14) Schur’s Theorem Let be a linear operator on a finite dimensional inner product space . Suppose that the characteristic polynomial of splits. Then there exists an orthonormal basis such that the matrix is upper triangular.
Intuition: will have an eigenvalue since splits. Let be the corresponding eigenvalue. Consider . It can be shown that is -invariant and therefore (Friedberg 5.26) applies to the characteristic polynomial of . Induction can then be used to prove the rest of the details.
Let be an inner product space and a linear operator on . is normal if
Similarly is normal if
(Friedberg 6.15) Let be an inner product space, and let be a normal operator on . Then
for all
is normal for every
If is an eigenvector of then is also an eigenvector of . In fact
If and are distinct eigenvalues of with corresponding eigenvectors and . Then and are orthogonal.
(Friedberg 6.16) Let be a linear operator on a finite-dimensional complex inner product space . Then is normal if and only if there exists an orthonormal basis of eigenvectors of .