Ex4 16

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.16}$

This exercise shows that for any Hermitian operator $O:V\to V$, the direct sum of all eigenspaces of $O$ is $V$.

A unitary operator $U$ satisfies $U^\dagger U = I$.

a) Show that the columns of a unitary matrix $U$ form an orthonormal set.

b) Show that if $O$ is Hermitian, then so is $UOU^{-1}$ for any unitary operator $U$.

c) Show that any operator has at least one eigenvalue $\lambda$ and $\lambda$-eigenvector $v_\lambda$.

d) Use part c) to show that for any matrix $A:V\to V$, there is a unitary operator $U$ such that the matrix for $UAU^{-1}$ is upper triangular (meaning all entries below the diagonal are zero).

e) Show that for any Hermitian operator $O:V\to V$ with eigenvalues $\lambda_1, \dots, \lambda_k$, the direct sum of the $\lambda_i$-eigenspaces $S_{\lambda_i}$ gives the whole space:

(1)
\begin{align} V = S_{\lambda_1}\oplus S_{\lambda_2}\oplus\cdots\oplus S_{\lambda_k}. \end{align}