Page 191 - Advanced Linear Algebra
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The Structure of a Linear Operator 175
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1 The multiset of elementary divisors or invariant factors is a complete
invariant for similarity of operators, that is,
¯ = =
¯ ³ ~ ² ElemDiv ² ElemDiv ³
¯ ³ ~ ² InvFact ² InvFact ³
A similar statement holds for matrices:
( ) ¯ - -
( )
¯ ² ElemDiv ( ³ ~ ² ElemDiv ) ³
¯ ² InvFact ( ³ ~ ² InvFact ) ³
2 The connection between operators and their representing matrices is
)
8 for some ¯ = -
(~ ´ µ 8 (
¯ ² ElemDiv ³ ~ ² ElemDiv ( ³
¯ ² InvFact ³ ~ ² InvFact ( ³
Theorem 7.13 can be summarized in Figure 7.1, which shows the big picture.
similarity classes
W V
of L(V)
isomorphism classes
V W V V of F[x]-modules
Multisets of
{ED } {ED } elementary divisors
1
2
[W] B [V] B Similarity classes
[W] [V] of matrices
R R
Figure 7.1
Figure 7.1 shows that the similarity classes of B²= ³ are in one-to-one
correspondence with the isomorphism classes of -´%µ -modules = and that these
are in one-to-one correspondence with the multisets of elementary divisors,
which, in turn, are in one-to-one correspondence with the similarity classes of
matrices.
We will see shortly that any multiset of prime power polynomials is the multiset
of elementary divisors for some operator (or matrix) and so the third family in
