Page 366 - A Course in Linear Algebra with Applications

350 Chapter Nine: Advanced Topics

a corresponding concept for matrices. The minimum poly-
nomial of a square matrix A over a field F is defined to be
the monic polynomial / with coefficients in F of least degree
such that f(A) = 0. The existence of / is assured by 9.4.1
and the relationship between linear operators and matrices.
Clearly the minimum polynomial of a linear operator equals
the minimum polynomial of any representing matrix. There
is of course an exact analog of 9.4.1 for matrices.
Example 9.4.1
What is the minimum polynomial of the following matrix?

/ 2 1 1
4 = I 0 2 0
\ 0 0 2

In the first place we can see directly that (A — 2J3) 2 = 0.
Therefore the minimum polynomial / must divide the poly-
2
nomial (x — 2) , and there are two possibilities, / = x — 2 and
2
f = (x — 2) . However / cannot equal x — 2 since A — 21 7^ 0.
2
Hence the minimum polynomial of A is / = (x — 2) .
Example 9.4.2
What is the minimum polynomial of a diagonal matrix D?

Let d\,. . ., d r be the distinct diagonal entries of D. Again
there is a fairly obvious polynomial equation that is satisfied
by the matrix, namely

(A - dj) •••(A- d rI) = 0.

So the minimum polynomial divides (x — d\) • • • (x — d r) and
hence is the product of certain of the factors x — di. However,
we cannot miss out even one of these factors; for the product
of all the A — djI for j ^ i is not zero since dj ^ di. It follows
that the minimum polynomial of D is the product of all the
factors, that is, (x — d\) • • • (x — d r).