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Section 5.9 Matrix Decompositions
UNDER CONSTRUCTION
Returning to our special matrices, we have shown that both Hermitian and unitary matrices admit orthonormal basis of eigenvectors. So suppose that
\begin{equation}
M |v\rangle = \lambda |v\rangle ,
M |w\rangle = \mu |w\rangle ,
..\text{.}\tag{5.9.1}
\end{equation}
where
\(|v\rangle\text{,}\) \(|w\rangle\text{,}\) ... satisfy
(5.7.1). Then
(5.7.3) is satisfied, and a similar argument shows that
\begin{equation}
M = \lambda |v \rangle \langle v| + \mu |w \rangle \langle w| + ... ~\text{,}\tag{5.9.2}
\end{equation}
so that we can expand any Hermitian or unitary matrix in terms of its eigenvalues and the projection operators formed from its eigenvectors.
