Linear Algebra

Section IV. Jordan Form 397
�
5 4
�
3
(d) −1 0 −3
1
�
9
−2
7
1
�
3
(e) −9 −7 −4
4
�
2
4
2
4
�
−1
(f) −1 −1 1
−1
⎛
7
−2
1
2
2
⎞
2
⎜ 1
(g) ⎝
−2
4
1
−1
5
−1⎟
−1
⎠
1 1 2 8
� 2.22 Find all possible Jordan forms of a transformation with characteristic polynomial
(x − 1) 2 (x +2) 2 .
2.23 Find all possible Jordan forms of a transformation with characteristic polynomial
(x − 1) 3 (x +2).
� 2.24 Find all possible Jordan forms of a transformation with characteristic polynomial
(x − 2) 3 (x + 1) and minimal polynomial (x − 2) 2 (x +1).
2.25 Find all possible Jordan forms of a transformation with characteristic polynomial
(x − 2) 4 (x + 1) and minimal polynomial (x − 2) 2 (x +1).
� 2.26 Diagonalize � � these. �
1 1
0
(a)
(b)
0 0
1
�
1
0
� 2.27 Find the Jordan matrix representing the differentiation operator on P3.
� 2.28 Decide if these two are similar.
� �
1 −1
�
−1
�
0
4 −3 1 −1
2.29 Find the Jordan form of this matrix.
� �
0 −1
1 0
Also give a Jordan basis.
2.30 How many similarity classes are there for 3×3 matrices whose only eigenvalues
are −3 and4?
� 2.31 Prove that a matrix is diagonalizable if and only if its minimal polynomial
has only linear factors.
2.32 Give an example of a linear transformation on a vector space that has no
non-trivial invariant subspaces.
2.33 Show that a subspace is t − λ1 invariant if and only if it is t − λ2 invariant.
2.34 Prove or disprove: two n×n matrices are similar if and only if they have the
same characteristic and minimal polynomials.
2.35 The trace of a square matrix is the sum of its diagonal entries.
(a) Find the formula for the characteristic polynomial of a 2×2 matrix.
(b) Show that trace is invariant under similarity, and so we can sensibly speak
of the ‘trace of a map’. (Hint: see the prior item.)
(c) Is trace invariant under matrix equivalence?