Given an orthogonal matrix, how can I find a second orthogonal matrix that gives a product of zero?

Question

Assume I have some orthogonal matrix $\mathbf{Q}^\text{T}\mathbf{Q} = \mathbf{I}$, How can I find a second orthogonal matrix $\mathbf{S}^\text{T}\mathbf{S} = \mathbf{I}$ that gives a product of zero, i.e

$$ \mathbf{Q}^\text{T}\mathbf{S} = \mathbf{S}^\text{T}\mathbf{Q} = \mathbf{0} $$ I've read about the null space matrix. If $\mathbf{S}$ is the null space matrix of $\mathbf{Q}$ and $\mathbf{Q}$ is orthonormal, then is $\mathbf{S}$ also orthonormal.

Answer 1

You can't: both matrices $Q$ and $S$ (and their transposes) are invertible, so their product is also invertible and hence can't be the zero matrix.

About the second question: The null space of $Q$ is $\{0\}$, so the null space matrix would be $[0]$ which is not orthonormal.

Answer 2

You can't. Any orthogonal matrix is nonsingular, and cannot have a product of zero with any nonzero matrix.

Answer 3

This isn't possible. The rows (columns) of $Q,S$ form an orthonormal basis for the space.

So, if you take any non-zero vector $s$, $Qs$ is non-zero in at least one coordinate (since you're writing $s$ in the basis given by $Q$'s rows). Now, apply this idea to $Q^T S$ or $S^T Q$ and you see its impossible.