Eigenvalue decomposition of a symmetric matrix
Let
We solve for the characteristic equation:
Hence the eigenvalues are , . For each eigenvalue , we look for a unit-norm vector such that . For , we obtain the equation in
which leads to (after normalization) an eigenvector . Similarly for we obtain the eigenvector . Hence, admits the SED (Symmetric Eigenvalue Decomposition)
See also: Sums-of-squares for a quadratic form.