Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $M_n(\mathbb{R})$?

share|improve this question
1  
Are you requiring that the positive definite matrices be symmetric? Some people don't require this, some do. –  mtiano Jan 21 at 14:46
2  
"Generate" as in "linear span"? –  user66081 Jan 21 at 14:50
    
Yes, I require them to be symmetric and yes, I meant linearly span. –  giulio Jan 21 at 15:04

1 Answer 1

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).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.