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Assume $A\in M_n(\mathbb{R})$ and $\det(A)\not=0$, is there existing two orthogonal matrix $O_1$,$O_2$ that satisfy $$O_1AO_2=\begin{pmatrix}\lambda_1 & & & \cr & \lambda_2 & & \cr & & \dots & \cr & & & \lambda_n\end{pmatrix}$$ where $\lambda_i>0$

how to prove?

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See SVD – chaohuang Nov 16 '12 at 17:21
up vote 1 down vote accepted

This is a special case of the Singular Value Decomposition.

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