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,…,vr spanning the image of A, and let w1,…,wr be preimages of v1,…,vr. Choose B mapping vi to wi 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) = xx2h(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.

About Peter Cameron

I count all the things that need to be counted.
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4 Responses to Something I didn’t know

  1. Mark C. Wilson says:

    1) This is interesting.
    2) I just saw your report on the conference in Auckland Dec 2008 that I organized. Some of the blog entries there might have been better conveyed to the organizer.
    3) The Oxford guide to English usage recommends -ize and not -ise for worlds like “diagonalize”. This is a battle I am getting a bit tired of fighting, but I think it is another thing you should know.

    • 1. Although I have been working on regular semigroups for more than ten years (where existence of generalised inverses is the definition), this question actually came from statistics, where the generalised inverse is used to extract information from data.
      2. I don’t remember writing a report on the 2008 Auckland conference, and Google won’t find anything like this.
      3. As I understand it, the rule is that Greek words take “ize” (so “diagonalize” is correct) but words that have entered English from French or Latin take “ise”. Living in St Andrews, I tend to take Chambers as the authority. Unfortunately Chambers gives “generalize” even though it admits the word comes from French.
      It is my belief that the idiosyncratic spelling of English is worth retaining since it gives us information about word origins — but not if we get it wrong!

    • Ah, so you are referring to the Quadrant. Well, I knew to expect trouble, I had had a run-in with them earlier the same year on the Forder tour. When we arrived, the desk clerk looked a bit embarrassed and said he needed to call the manager. The manager, who was little if any older than the desk clerk, explained that despite the booking they didn’t have a room for us, and had put us in a hotel in Viaduct Harbour. I asked if you could walk to the University from there, and was assured that it was quite impossible.
      We stayed in the Quadrant again when we were in Auckland on Hood fellowships. That time, nothing went wrong (except for the washing machine shrinking our clothes).
      I don’t think it was really anything for which the conference organisers need take responsibility.

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