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Is it true that positive definite matrices generates all the symmetric matrices in $M_n(\mathbb{R})$?

And is it true that the set of nonsingular symmetric matrices generates all the symmetric matrices $M_n(\mathbb{R})$?

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Are you requiring that the positive definite matrices be symmetric? Some people don't require this, some do. – Wintermute Jan 21 '14 at 14:46
"Generate" as in "linear span"? – user66081 Jan 21 '14 at 14:50
Yes, I require them to be symmetric and yes, I meant linearly span. – giulio Jan 21 '14 at 15:04

If "generate" means "span linearly" then this might help:

Any symmetric real matrix $A$ is of the form $U^T D U$, where $U$ is orthogonal, and $D$ is diagonal and real, and vice versa.

Substitute $D = (g I + D) - g I$, where $g$ is a big positive number.

You can do this on the original matrix also, but it is slightly less evident that $A + g I$ is positive definite. You can use the Gerschgorin disk theorem, or check directly that $x^T (A + g I) x$ is indeed positive for all nonzero $x$.

So the answer is yes, to both questions.

If "generate" means "by matrix product" then:

No: consider the zero matrix (proof by the determinant-of-product-is-product-of-determinant formula).

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