Page 179 - Advanced Linear Algebra

Chapter 7

The Structure of a Linear Operator

In this chapter, we study the structure of a linear operator on a finite-
dimensional vector space, using the powerful module decomposition theorems
of the previous chapter. Unless otherwise noted, all vector spaces will be
assumed to be finite-dimensional.

Let be a finite-dimensional vector space. Let us recall two earler theorems
=
(Theorem 2.19 and Theorem 2.20).
Theorem 7.1 Let be a vector space of dimension .
=

(
)
)
(
1 Two d matrices and are similar written ( — ) ) if and only if
they represent the same linear operator B ²= ³ , but possibly with
respect to different ordered bases. In this case, the matrices and )
(
represent exactly the same set of linear operators in B²= ³ .
)
=

2 Then two linear operators and on are similar written — ( ) if and
that represents both operators, but with
only if there is a matrix ( C
respect to possibly different ordered bases. In this case, and are

represented by exactly the same set of matrices in C .
Theorem 7.1 implies that the matrices that represent a given linear operator are
precisely the matrices that lie in one similarity class. Hence, in order to uniquely
represent all linear operators on , we would like to find a set consisting of one
=
simple representative of each similarity class, that is, a set of simple canonical
forms for similarity.
One of the simplest types of matrix is the diagonal matrix. However, these are
too simple, since some operators cannot be represented by a diagonal matrix. A
less simple type of matrix is the upper triangular matrix. However, these are not
simple enough: Every operator (over an algebraically closed field) can be
represented by an upper triangular matrix but some operators can be represented
by more than one upper triangular matrix.