Linear Algebra, 2020a

448 Chapter Five. Similarity
(c) Verify the statement, excerpted here, that precedes Theorem 1.8.
. . . if a minimal polynomial m(x) for a transformation t factors as
m(x) =(x − λ 1 ) q 1 ···(x − λ z ) q z
then m(t) =(t − λ 1 ) q 1 ◦···◦(t − λ z ) q z
is the zero map. Since m(t) sends every vector to zero, at least one of
the maps t − λ i sends some nonzero vectors to zero. . . . That is, . . .
at least some of the λ i are eigenvalues.
1.30 True or false: for a transformation on an n dimensional space, if the minimal
polynomial has degree n then the map is diagonalizable.
1.31 Let f(x) be a polynomial. Prove that if A and B are similar matrices then f(A)
is similar to f(B).
(a) Now show that similar matrices have the same characteristic polynomial.
(b) Show that similar matrices have the same minimal polynomial.
(c) Decide if these are similar.
(1 ) ( )
3 4 −1
2 3 1 1
1.32 (a) Show that a matrix is invertible if and only if the constant term in its
minimal polynomial is not 0.
(b) Show that if a square matrix T is not invertible then there is a nonzero matrix
S such that ST and TS both equal the zero matrix.
̌ 1.33 (a) Finish the proof of Lemma 1.7.
(b) Give an example to show that the result does not hold if t is not linear.
1.34 Any transformation or square matrix has a minimal polynomial. Does the
converse hold?
IV.2
Jordan Canonical Form
We are looking for a canonical form for matrix similarity. This subsection
completes this program by moving from the canonical form for the classes of
nilpotent matrices to the canonical form for all classes.
2.1 Lemma A linear transformation on a nontrivial vector space is nilpotent if
and only if its only eigenvalue is zero.
Proof If the linear transformation t on a nontrivial vector space is nilpotent
then there is an n such that t n is the zero map, so t satisfies the polynomial
p(x) =x n =(x − 0) n . By Lemma 1.10 the minimal polynomial of t divides p, so
the minimal polynomial’s only root is zero. By Cayley-Hamilton, Theorem 1.8,
the characteristic polynomial’s only root is zero. Thus t’s only eigenvalue is
zero.