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Is it possible to find a close symmetric matrix (positive definite or not) to every non-singular matrix? What I mean is that given some nonsingular matrix A and some $\epsilon > 0$, can I find a symmetric $\tilde{A}$ such that $||A-\tilde{A}||_F < \epsilon$? What happens to this question when I look at constrained cases: like matrices with unit determinant det(A)=1?

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It's certainly not true for arbitrary matrices and values of $\epsilon$.

For example, if you set $$A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$$

and $\epsilon = 10^{-40}$ then the problem has no solution.

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