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Spectral Decomposition.

Prove that if A is symmetric, and orthogonally diagonalized by P = [u1 · · · un],

then

$A =\sum_{k=1}^n \lambda_k \, u_k \,u_k^T$

where the $\lambda_k$ are the eigenvalues of $A$.

Hint:

If $A = P DP^T$ where $P$ (as given above) is orthogonal and $D = (\lambda_i \delta_{ij})$, use the definition of matrix multiplication, the fact that $D$ is diagonal, the orthonormality of the $\{u_k\}_{k=1\dots n}$, and the following formula: if $A = XY Z$, where all are $n \times n$ matrices, then

$a_{ij} = \sum_{k=1}^n \sum_{l=1} x_{ik}\, y_{kl} \, z_{lj}, \forall i, j = 1, \dots n.$

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I think it's much easier. What is $Au_m$ equal to?

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