Something I didn’t know

I didn’t know this, though probably I should have. Maybe you didn’t know it either.

We work in a semigroup, a system with an operation (called multiplication) satisfying the associative law. A generalised inverse of an element A is an element B satisfying ABA = A. The name comes from the fact that, if there is an identity element I, then an inverse of A (an element B satisfying AB = I) is a generalised inverse.

If a generalised inverse B of A exists, then we may assume that A is also a generalised inverse of B, that is, BAB = B. To see this, put C = BAB. Then

• ACA = ABABA = ABA = A,
• CAC = BABABAB = BABAB = BAB = C.

So C is also a generalised inverse of A and has the required property.

Now we come to the bit that I didn’t know. Let us consider matrices, over an arbitrary field. The following two theorems hold:

Theorem 1 Every matrix has a generalised inverse.

For let A be a matrix. Choose vectors v₁,…,v_r spanning the image of A, and let w₁,…,w_r be preimages of v₁,…,v_r. Choose B mapping v_i to w_i for i = 1,…,r. Then ABA = A.

Theorem 2 For a matrix A, the following are equivalent:

1. A has a generalised inverse which commutes with A;
2. A has a generalised inverse which is a polynomial in A;
3. 0 is not a repeated root of the minimal polynomial of A.

Here is the proof.

2 implies 1: Clear.

3 implies 2: Suppose that 0 is not a repeated root of the minimal polynomial of A. Then there is a polynomial f with zero constant term and non-zero coefficient of x which is satisfied by A. (If 0 is a root of the minimal polynomial, use this; otherwise use the minimal polynomial multiplied by x.) After multiplying by a non-zero scalar we can write f(x) = x−x²h(x). Then h(A) is a generalised inverse of A.

1 implies 3: Suppose that ABA = A and AB = BA. Then BA² = A. But, if 0 is a repeated root of the minimal polynomial of A, then there is a vector v which is mapped to 0 by A² but not by A, and applying the above equation to v gives a contradiction.

Corollary If a matrix A is diagonalisable, then it has a generalised inverse which is a polynomial in A. In particular, this holds for real symmetric matrices.