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