Linear Algebra

Section IV. Jordan Form 385
� 1.17 Find the minimal polynomial of matrices of this form
⎛
⎞
λ 0 0 ... 0
⎜1
⎜
⎜0
⎜
⎝
λ
1
0
λ
. ..
λ
0 ⎟
0⎠
0 0 ... 1 λ
where the scalar λ is fixed (i.e., is not a variable).
1.18 What is the minimal polynomial of the transformation of Pn that sends p(x)
to p(x +1)?
1.19 What is the minimal polynomial of the map π : C 3 → C 3 projecting onto the
first two coordinates?
1.20 Find a 3×3 matrix whose minimal polynomial is x 2 .
1.21 What is wrong with this claimed proof of Lemma 1.9: “ifc(x) =|T −xI| then
c(T )=|T − TI| = 0”?
1.22 Verify Lemma 1.9 for 2×2 matrices by direct calculation.
� 1.23 Prove that the minimal polynomial of an n × n matrix has degree at most
n (not n 2 as might be guessed from this subsection’s opening). Verify that this
maximum, n, can happen.
� 1.24 The only eigenvalue of a nilpotent map is zero. Show that the converse statement
holds.
1.25 What is the minimal polynomial of a zero map or matrix? Of an identity map
or matrix?
� 1.26 Interpret the minimal polynomial of Example 1.2 geometrically.
1.27 What is the minimal polynomial of a diagonal matrix?
� 1.28 A projection is any transformation t such that t 2 = t. (For instance, the
transformation of the plane R 2 projecting each vector onto its first coordinate will,
if done twice, result in the same value as if it is done just once.) What is the
minimal polynomial of a projection?
1.29 The first two items of this question are review.
(a) Prove that the composition of one-to-one maps is one-to-one.
(b) Prove that if a linear map is not one-to-one then at least one nonzero vector
from the domain is sent to the zero vector in the codomain.
(c) Verify the statement, excerpted here, that preceeds Theorem 1.8.
... if a minimial polynomial m(x) for a transformation t factors as
m(x) =(x − λ1) q1 ···(x − λℓ) q ℓ then m(t) =(t − λ1) q1 ◦···◦(t − λℓ) q ℓ
is the zero map. Since m(t) sends every vector to zero, at least one
of the maps t − λi sends some nonzero vectors to zero. ... Rewording
... : at least some of the λi 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.