I have to find an example of a matrix $Q$ that has orthonormal columns, but $QQ^T \neq I$.
If a matrix has orthonormal columns, it does not imply that the matrix is orthogonal, so that it is a square matrix. Therefore, I could simply give a matrix with unit vectors that is not a square matrix, since the $QQ^T$ would not be a identity, right?
Best Answer
Your idea is right, although you would still need to have some verification that $QQ^T$ is not, in fact, the identity. It's certainly conceivable (without some theorem to back you up…) that you could get unlucky and choose a non-square $Q$ for which $QQ^T$ was still the identity matrix.