Unformatted text preview: 410 8. MODULES Deﬁnition 8.7.3. A matrix is said to be in Jordan canonical form if it is
block diagonal with Jordan blocks on the diagonal.
Jm1 . 1 /
Jm2 . 2 /
0 0 Jm t . t / Theorem 8.7.4. (Jordan canonical form for a linear transformation.) Let
T be a linear transformation of a ﬁnite dimensional vector space V over a
ﬁeld K . Suppose that the characteristic polynomial of T factors into linear
factors in KŒx.
(a) V has a basis with respect to which the matrix of T is in Jordan
(b) The matrix of T in Jordan canonical form is unique, up to permutation of the Jordan blocks. Proof. We have already shown existence of a basis B such that ŒT B is in
Jordan canonical form. The proof of uniqueness amounts to showing that
a matrix in Jordan canonical form determines the elementary divisors of
T . In fact, suppose that the matrix A of T with respect to some basis is in
Jordan canonical form, with blocks Jmi . i / for 1 Ä i Ä t . It follows that
.T; V / has a direct sum decomposition
.T; V / D .T1 ; V1 / ˚ ˚ .T t ; V t /; where the matrix of Ti with respect to some basis of Vi is Jmi . i /. By
Lemma 8.7.2, Vi is a cyclic KŒx–module with period .x
i / . By
uniqueness of the elementary divisor decomposition of V , Theorem 8.5.16,
the polynomials .x i /mi are the elementary divisors of the KŒx–module
V , that is, the elementary divisors of T . Thus, the blocks of A are uniquely
determined by T , up to permutation.
I Deﬁnition 8.7.5. A matrix is called the Jordan canonical form of T if
it is in Jordan canonical form, and
it is the matrix of T with respect to some basis of V .
The Jordan canonical form of T is determined only up to permutation of
its Jordan blocks. ...
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- Fall '08
- Algebra, Jordan Canonical Form