# Singular Value Decomposition

Algebra Level pending

Let $A=\left(\begin{array}{ccc} 5&-1&2\\ -1&5&2\end{array}\right).$ Then, by singular value decomposition, there is a diagonal matrix $$\Sigma$$ and orthogonal matrices $$U$$ and $$V$$ such that $$A=U\Sigma V$$. The sum of the entries on the main diagonal of $$\Sigma$$ can be written in the form $$a+b\sqrt{c}$$, where $$a,b,c$$ are positive integers and $$c$$ square-free. Find $$a+b+c$$.

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