Thread: Is sum of positive definite matrices positive definite

Is sum of two positive definite matrices positive definite and is there a
proof or a theorem that shows that?

Thank you.

Originally Posted by itpro

Is sum of two positive definite matrices positive definite and is there a proof or a theorem that shows that?

Yes, if are positive definite then, for all . This implies for all .

This is great, Thank you!

Is there a similar proof for products of two positive definite matrices?

Originally Posted by itpro

Is sum of two positive definite matrices positive definite and is there a
proof or a theorem that shows that?

Thank you.

A hint (I belive) :

If the matrix is positive definite, then all its eigenvalues are strictly positive.

Originally Posted by itpro

This is great, Thank you! Is there a similar proof for products of two positive definite matrices?

No, there isn't. The product of two definite positive matrices is definite positive if and only if the matrices commute.