Singular Value Decomposition

Algebra Level 4

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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