Relationship between cross-product and Moore-Penrose pseudoinverse

Question

It is said in here https://blog.bham.ac.uk/intellimic/g-landini-software/colour-deconvolution-2/ that you can get the third vector of a 3x3 (stain) matrix either by taking the cross product of the first two vectors (rows) of the (stain) matrix or by taking the Moore-Penrose pseudoinverse of that (stain) matrix. In other words, that cross product and Moore-Penrose pseudoinverse are somehow the same ("using the crossed product...is equivalent to resolving the problem using the Moore-Penrose pseudoinverse"). How? I have never seen Moore-Penrose pseudoinverse used in this context (only to find a pseudoinverse of a matrix that is not invertible, not to find a cross product). Does anyone know why this is the case? How do you get from a pseudoinverse to a cross-product? PS: The article discusses how to mathematically unmix stains in digital images of biological tissues stained for the purposes of their histochemical analysis. Hence, the name "stain" matrix.

Answer 1

In this case, the cross product is used to get a non-zero vector that is orthogonal to two known vectors $v_1,v_2$. In other words, we're looking for a non-zero solution to the equation $Ax = 0$, where $A$ is the matrix whose rows are $v_1,v_2$.

As is noted on the Wikipedia article, the solutions to an equation of the form $Ax = b$ are of the form $x = A^+b + [I - A^+A]w$ (where $A^+$ denotes the pseudoinverse of $A$). In our case, we have $b = 0$, which means that our solutions are of the form $x = [I - A^+A]w$.

Notably, $I - A^+A$ is a non-zero matrix. Taking $x$ to be any non-zero column of $I - A^+A$ (corresponding to choosing $w$ to be one of the standard basis vectors) gives us the desired result.