Page 163 - DMTH502_LINEAR_ALGEBRA

Unit 12: Introduction and Characteristic Values of Elementary Canonical Forms

Since c is a characteristic value of A if and only if det (A – cI) = 0, we form the matrix (xI – A) with Notes
polynomial entries, and consider the polynomial f = det (xI – A). Clearly the characteristic
values of A in F are just the scalars c in F such that f(c) = 0. For this reason f is called the
characteristic polynomial of A. It is important to note that f is a monic polynomial which has
degree exactly n. This is easily seen from the formula for the determinant of a matrix in terms of
its entries.
Lemma: Similar matrices have the same characteristic polynomial.

Proof: If B = P AP, then
–1
det (xI – B) = det (xI – P PA)
–1
= det (P (xI – A)P)
–1
–1
= det P . det (xI – A) . det P
= det (xI – A)
This lemma enables us to define sensibly the characteristic polynomial of the operator T as the
characteristic polynomial of any n × n matrix which represents T in some ordered basis for V.
Just as for matrices, the characteristic values of T will be the roots of the characteristic polynomial
for T. In particular, this shows us that T cannot have more than n distinct characteristic values. It
is important to point out that T may not have any characteristic values.

2
Example 1: Let T be the linear operator on R which is represented in the standard
ordered basis by the matrix
0 1
A =
1 0
The characteristic polynomial for T (or for A) is
x 1
det (xI – A) = = x + 1.
2
1 x
Since this polynomial has no real roots, T has no characteristic values. If U is the linear operator
2
on C which is represented by A in the standard ordered basis, then U has two characteristic
value, i and –i. Here we see a subtle point. In discussing the characteristic values of a matrix A, we
must be careful to stipulate the field involved. The matrix A above has no characteristic values
in R, but has the two characteristic values, i and –i in C.

Example 2: Let A be the (real) 3 × 3 matrix

3 1 1
2 2 1
2 2 0

Then the characteristic polynomial for A is
x 3 1 1
2
2 x 2 1 x 3 5x 2 8x 4 (x 1)(x 2) .
2 2 x
Thus the characteristic values of A are 1 and 2.

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