Given $ A $ find $ B $ so that $ [ A \: B ] [ A \: B ] ^ { \mathrm T } $ is the identity matrix

Given $ A $ ($ 4 \times 2 $ matrix) I want to find $ B $ ($ 4 \times 2 $ matrix) such that when you append both in a single $ 4 \times 4 $ matrix $ [ A \: B ] $ the following relationship will hold: $$ [ A \: B ] [ A \: B ] ^ { \mathrm T } = I $$

Edit context:
I am going through a paper on unambiguous state discrimination where a process has been outlined for finding optimal discrimination matrices. This is the paper https://ieeexplore.ieee.org/document/4557236. In section (3) the paper uses the matrix I have called $ B $ but does not however tell how to get the matrix. $ A $ is made up of the null space vectors of a rank $ 2 $ matrix in a $ 4 $-dimensional Hilbert space. $ A $ is also real.

Solution 1:

I cannot access the paper you linked. However, I suspect that "null space vectors" refer to an orthonormal basis for the nullspace.

If this is the case, then the columns of $A$ are orthonormal, and it follows that the matrix $M = [A\ \ B]$ will satisfy $MM^T = I$ if and only if $M^TM = I$, which holds if and only all columns of $M$ are orthonormal. Thus, a satisfactory matrix $B$ can be build from the nullspace vectors of $A^T$.