Spectral decomposition of a symmetric matrix

Let us denote as A an N x N diagonal matrix, formed by the eigenvalues of the symmetric N x N matrix A, 

Multiplying both sides of equation (E.IO) by and taking into account (E.8), we obtain 

Formulae (E.ll) and (E.12) present a spectral decomposition of a matrix A. Note that from equation (E.ll) it follows immediately that 

The spectral decomposition can be effectively used for matrix inversion, taking into account property (E.8) of the eigenvector matrix  

Some formulae and rules from matrix algebra 

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The spectral decomposition of a symmetric matrix A can be rewritten in the form ... 

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