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 v_{1},…,v_{r} spanning the image of A, and let w_{1},…,w_{r} be preimages of v_{1},…,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:
- A has a generalised inverse which commutes with A;
- A has a generalised inverse which is a polynomial in A;
- 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−x^{2}h(x). Then h(A) is a generalised inverse of A.
1 implies 3: Suppose that ABA = A and AB = BA. Then BA^{2} = 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 A^{2} 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.