Singular Value Decomposition

Algebra Level 4

Let A=(512152).A=\left(\begin{array}{ccc} 5&-1&2\\ -1&5&2\end{array}\right). Then, by singular value decomposition, there is a diagonal matrix Σ\Sigma and orthogonal matrices UU and VV such that A=UΣVA=U\Sigma V. The sum of the entries on the main diagonal of Σ\Sigma can be written in the form a+bca+b\sqrt{c}, where a,b,ca,b,c are positive integers and cc square-free. Find a+b+ca+b+c.

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