# When is the gram matrix positive definite?

1. Dec 9, 2015

### perplexabot

Hey all. I know that $A^TA$ is positive semidefinite. Is it possible to achieve a positive definite matrix from such a matrix multiplication (taking into account that $A$ is NOT necessarily a square matrix)?

2. Dec 13, 2015

### Hawkeye18

$A^TA$ is positive definite if and only if the columns of $A$ are linearly independent (equivalently, $A$ is left invertible).

PS. Of course, that is only true for real matrices, for complex matrices you need to replace $A^TA$ by $A^*A$ where $A^*A$ is the Hermitian adjoint of $A$.

3. Dec 13, 2015

### perplexabot

Wow, thank you.