Orthonormal matrix whose column space is orthogonal to another matrix.

Question

I am reading a paper on controlling false discovery rate for variable selection.

This paper constructs Knockoff matrix $\tilde{X}$ for original design $X$ using equation

$$ \tilde{X} = X\left ( I -\Sigma ^{-1}diag\left \{ s \right \}\right ) + \tilde{U}C $$ here, $\tilde{U}$ is an $n × p$ orthonormal matrix whose column space is orthogonal to that of $X$ so that $\tilde{U}^{T}X = 0$, and $C^{T}C = 2diag\left \{ s \right \} - diag\left \{s \right \} \Sigma ^{-1}diag\left \{ s \right \}$ is a Cholesky decomposition

I am having trouble understanding how to construct an orthonormal matrix whose column space is orthogonal to an another matrix.

Please help.

Barber, Rina Foygel; Candès, Emmanuel J., Controlling the false discovery rate via knockoffs, Ann. Stat. 43, No. 5, 2055-2085 (2015). ZBL1327.62082.>

Answer 1

If "orthonormal matrix" means a matrix whose columns are orthonormal (the terminology is not quite standard), you can take $p$ linearly independent vectors in the orthogonal complement of the column space of $X$ and use Gram-Schmidt to get orthonormal vectors which still have that property, then put them together as columns to form $U$.