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Commuting matrices as polynomials?
Posted:
Feb 18, 2008 2:44 PM
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Suppose A and B are commuting nxn matrices. It does not follow that
one of these can be expressed as a polynomial in the other. It also
does not follow that A and B can both be expressed as polynomials in
terms of a third matrix C.
In a book by Marcus and Minc (Survey of Matrix Theory and Matrix
Inequalities) they give the matrices A=[ [0 0 0], [2 0 0], [0 0 0]]
and B=[[0 0 0],[0 0 3], [0 0 4]] as a example (these are given row by
row) of the second statement. That is, AB=0 and BA=0 so they commute
and Marcus & Minc say that there is no C such that A=p(C) and B=q(C)
for polynomials p and q.
A bit of hand computation led me to
C=[[0 0 0], [1 0 3], [0 0 4]] giving B =1/4 C^2 and A=2C-1/2 C^2
Unless I'm missing something it seems that the example given is
incorrect.
(1) What is the example supposed to be?
(2) Can anyone give me a reference to a source that discusses
statement 2 (i.e. what is required for A and B to be expressible as
polynomials in a third matrix C). I have quite a few linear algebra
books but none discuss this)
Thanks
jw
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