# Positive Definite Matrix Multiplication Non-Symmetric

If we have $$B$$ is an $$(n,n+1)$$ matrix and $$A$$ is a positive definite $$(n,n)$$ matrix is the resulting quantity positive definite?

$$B^TAB$$

It seems like it should be since we have both a $$B^T$$ and a $$B$$ in there and we know that $$B^TB$$ is positive definite, but I am getting stuck on an explicit proof.

Thanks!

## 1 Answer

It is never positive definite. $$B$$ is a fat matrix. Thus $$Bx=0$$ has a nontrivial solution $$x$$ and $$x^TB^TABx=0$$.

• Hmm what is a fat matrix? Jul 13 '19 at 15:47
• Do all fat matrices have a guaranteed solution? Jul 13 '19 at 15:50
• @user2879934 When the number of columns exceeds the dimension of the vector space, the columns must be linearly dependent. Jul 13 '19 at 15:51
• What about semi-positive definite? Jul 13 '19 at 16:05
• Interesting, so if both A and B are semi-positive definite, does that rearrangement guarantee that quantity is >= 0. Sorry B can't be PSD Jul 13 '19 at 16:12