Maximum number of Q-orthogonal vectors

Question

Let $Q$ be a $n\times n$ positive semi-definite matrix, then $\{d_0, …, d_k\}$ is the set of "Q-orthogonal vectors" if and only if for all $d_i, d_j$, $d_i^TQd_j = 0$.
This is used in the Conjugate gradient method. I showed that all Q-orthogonal vectors are linearly independent, and I guess that if there are $n$ Q-orthogonal vectors that are linearly independent, then Q must be positive definite. Is my guess true? I couldn't find a way to prove it.

Answer 1

and I guess that if there are n Q-orthogonal vectors that are linearly independent, then Q must be positive definite.

This statement is false. Consider the case $Q = 0$. Then every pair of vectors is $Q$-orthogonal and so clearly there exists $n$ linearly independent $Q$-orthogonal vectors but $Q$ is not positive definite.