= Spectral projection of a self-adjoint matrix
{title2=$P_\lambda=\sum_{\lambda_j=\lambda}e_je_j^*$}
= Spectral projections of a self-adjoint matrix
{synonym}
For a finite-dimensional <Hermitian matrix> $A$, its spectral projection $P_\lambda$ is the <orthogonal projection> onto the <eigenspace> for $\lambda$. In an <orthonormal eigenbasis>, the displayed sum is over every basis <vector> with that <eigenvalue>. The projection is independent of the choice of basis within a repeated <eigenspace>. For distinct <eigenvalues> $\lambda,\mu$,
$$
P_\lambda^*=P_\lambda,\qquad P_\lambda^2=P_\lambda,\qquad
P_\lambda P_\mu=0,\qquad
\sum_\lambda P_\lambda=I,\qquad A=\sum_\lambda\lambda P_\lambda.
$$
This is the <spectral decomposition> grouped by distinct <eigenvalues>: decompose a <vector> into orthogonal pieces, then scale each piece. For the matrix $A=\begin{pmatrix}2&1\\1&2\end{pmatrix}$,
$$
P_3=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad
P_1=\frac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix},\qquad A=3P_3+P_1.
$$
Thus $P_3(1,0)^T=(1/2,1/2)^T$ and $P_1(1,0)^T=(1/2,-1/2)^T$: the two pieces add back to $(1,0)^T$. Squaring the matrix simply squares each scalar, $A^2=9P_3+P_1$, and similarly the <matrix exponential> is $e^{tA}=e^{3t}P_3+e^tP_1$. <Spectral projections of a self-adjoint matrix> turn repeated <matrix> calculations into <scalar> calculations on fixed <vector subspaces>.
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