Condition on $X$ so that $X^TX$ is positive definite

Question

If I have $X_{n \times {m}}$ matrix, where:

• $n \ne m$ in general case
• $\mathsf {rk}(X)=max(n,m)$

how can I define $X$ so that the $X^TX$ is positive definite (all eigenvalues positive). This covariance matrix should be $m \times m$.

In other words what should be condition for $X$ so that covariance matrix $X^TX$ is positive definite.

Answer 1

The condition is that the rank be equal to $m.$