From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Solutions to a matrix's Characteristic Function
Date: 5 Feb 1996 16:17:46 GMT
In article <4f30lo$q01@news.ios.com>,
Martin Stern wrote:
>We all know that a square matrix satisfies its characteristic function.
>Does anyone know if other matrix solutions to the characteristic function
>have any interesting properties (besides the N obvious solutions that are
>eigenvalue multiples of the identity matrix)?
This would be an interesting question (and indeed is essentially the
observation with which one begins Galois theory) except that the ring of
matrices lacks two characteristics shared by all fields:
(a) If AB=0 then A=0 or B=0
(b) For all A and B, AB=BA
These interfere with what you think will happen when you solve polynomial
equations in this ring.
Consider for example the 2 x 2 diagonal matrix A = diag(1, -1). Its
characteristic (and minimal) polynomial is X^2-1=0, and indeed
A^2-1=0. It is tempting to let B be another solution of this polynomial,
say B = -A, and consider the polynomial (X-A)(X-B).
Unfortunately, there are solutions to X^2-1=0 which are different from
both A and B. If we take X to be the identity matrix, then surely
X^2-1=0. However, just because (I-A)(I-B)=0 does not mean that I=A or
I=B, since property (a) does not hold among matrices. Indeed, we
could use for B the negative of the matrix A; then
I^2-I=(I-A)(I-B) really is zero but neither I-A nor I-B is zero.
Another example indicates a worse problem. Consider X=matrix([0,1],[1,0])
(the permutation matrix). This matrix has X^2=I, too. However, when we
compute (X-A)(X-B) we don't even get X^2-I -- rather, we get
X^2-AX-XB+AB = X^2- (AX-XA) -I, which for this particular X is the
_nonzero_ matrix -(AX-XA) = matrix([0,-2],[2,0]). Here it is the failure
of axiom (b) which interferes.
Really all you can say, if you know that X satisfies the characteristic
polynomial of A, is that the eigenvalues of X are among those of A,
and that the geometric multiplicity of each eigenvalue of X is at most
the (algebraic) multiplicity of that eigenvalue in the characteristic
polynomial of A. (Thus e.g. if A = diag(1, 1, -1) then X1 satisfies
the characteristic polynomial of A but X2 doesn't, where X1 and X2
each have X_ij=0 except that X_ii=1 for all i and X_12=1, and X2 has
an additional nonzero entry (X2)_23=1.)
dave