Linear Algebra, 2020a

462 Chapter Five. Similarity
̌ 2.22 Find all possible Jordan forms of a transformation with characteristic polynomial
(x − 2) 3 (x + 1) and minimal polynomial (x − 2) 2 (x + 1).
2.23 Find all possible Jordan forms of a transformation with characteristic polynomial
(x − 2) 4 (x + 1) and minimal polynomial (x − 2) 2 (x + 1).
̌ 2.24 Diagonalize
( )
these.
( )
1 1 0 1
(a)
(b)
0 0 1 0
̌ 2.25 Find the Jordan matrix representing the differentiation operator on P 3 .
̌ 2.26 Decide if these two are similar.
( ) ( )
1 −1 −1 0
4 −3 1 −1
2.27 Find the Jordan form of this matrix.
( ) 0 −1
1 0
Also give a Jordan basis.
2.28 How many similarity classes are there for 3×3 matrices whose only eigenvalues
are −3 and 4?
̌ 2.29 Prove that a matrix is diagonalizable if and only if its minimal polynomial has
only linear factors.
2.30 Give an example of a linear transformation on a vector space that has no
non-trivial invariant subspaces.
2.31 Show that a subspace is t − λ 1 invariant if and only if it is t − λ 2 invariant.
2.32 Prove or disprove: two n×n matrices are similar if and only if they have the
same characteristic and minimal polynomials.
2.33 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?
(d) Show that the trace of a map is the sum of its eigenvalues (counting multiplicities).
(e) Show that the trace of a nilpotent map is zero. Does the converse hold?
2.34 To use Definition 2.10 to check whether a subspace is t invariant, we seemingly
have to check all of the infinitely many vectors in a (nontrivial) subspace to see if
they satisfy the condition. Prove that a subspace is t invariant if and only if its
subbasis has the property that for all of its elements, t(⃗β) is in the subspace.
̌ 2.35 Is t invariance preserved under intersection? Under union? Complementation?
Sums of subspaces?
2.36 Give a way to order the Jordan blocks if some of the eigenvalues are complex
numbers. That is, suggest a reasonable ordering for the complex numbers.
2.37 Let P j (R) be the vector space over the reals of degree j polynomials. Show
that if j k then P j (R) is an invariant subspace of P k (R) under the differentiation
operator. In P 7 (R), doesanyofP 0 (R), ..., P 6 (R) have an invariant complement?