= Spectral projection of a diagonalizable matrix
{title2=$P_\lambda=\prod_{\mu\ne\lambda}(A-\mu I)/(\lambda-\mu)$}
= Spectral projections of a diagonalizable matrix
{synonym}
For a <diagonalizable matrix> with distinct <eigenvalues> $\lambda,\mu,\ldots$, this polynomial is the <projection operator> onto the <eigenspace> for $\lambda$ along the direct sum of the other <eigenspaces>. It is one on its own <eigenspace> and zero on the others. Consequently $P_\lambda P_\mu=0$ for different <eigenvalues>, $\sum P_\lambda=I$ and $A=\sum\lambda P_\lambda$. These projections need not be <orthogonal projections>; that stronger property holds for a <normal matrix>. This is the <polynomial interpolation> form of the <spectral decomposition>.
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