pinv broken on column-rank-deficient matrices.

[ixchow]

Applies to mathjs 14.1.0, installed via npm.

May be related to #3012, but that bug does not provide a test case.

According to wikipedia's summary, the pseudo-inverse of a column-rank-deficient matrix mat exists, and acts as a right inverse mat * pinv(mat) = I.

This does not appear to be the case when using mathjs.pinv on matricies with columns that are not linearly independent.

In some cases, the returned matrix is not a right inverse:

const mat = [
	[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
	[ 0, 0, 1, 1, 0, 0, 0, 0, 0, 1 ],
	[ 0, 0, 2, 0, 1, 0, 0, 1, 0, 0 ],
	[ 1, 0, 1, 0, 0, 0, 0, 0, 1, 0 ],
	[ 0, 1, 1, 0, 1, 1, 2, 0, 0, 1 ]
];

const pinv = math.pinv(mat);
console.table(math.multiply(mat, pinv));

The output here has its first column as all zeros -- quite far from the identity matrix (expected result).

In comparison, computing with wikipedia's summary of how to compute the pseudoinverse works just fine:

const mat = /* same as above */;
//right-inverse recipe, works for matrices with linearly-independent rows:
const pinv = math.multiply(
	math.transpose( mat ),
	math.inv(
		math.multiply(
			mat,
			math.transpose( mat )
		)
	)
);
console.table(math.multiply(mat, pinv));

Here, the output is very close to the identity matrix.

---

Further, pinv just fails on some matrices:

const mat = [
	[ 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
	[ 0, 0, 1, 1, 0, 0, 0, 0, 1 ],
	[ 0, 0, 0, 0, 1, 0, 1, 0, 0 ],
	[ 1, 0, 1, 0, 0, 0, 0, 1, 0 ],
	[ 0, 1, 1, 0, 1, 2, 0, 0, 1 ]
];
const pinv = math.pinv(mat);

Results in an Error: Cannot calculate inverse, determinant is zero exception being thrown. (The right inverse recipe above still works on this matrix.)

---

My suspicion is that the pinv implementation here was only tested on matrices with linearly-independent columns (the rank-surplus case), not linearly-independent rows (the rank-deficient case). It's a useful tool in both situations so it would make sense to patch to deal with both.

[gwhitney]

Thanks! This is a good addition to #3012. Having specific examples is very helpful. You don't happen to have an intuition as how we might find even smaller examples, do you?