Linear Algebra

394 Chapter Five. Similarity̌ 2.31 Prove that a matrix is diagonalizable if and only if its minimal polynomialhas only linear factors.2.32 Give an example of a linear transformation on a vector space that has nonon-trivial invariant subspaces.2.33 Show that a subspace is t − λ 1 invariant if and only if it is t − λ 2 invariant.2.34 Prove or disprove: two n×n matrices are similar if and only if they have thesame 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 speakof 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?2.36 To use Definition 2.6 to check whether a subspace is t invariant, we seeminglyhave to check all of the infinitely many vectors in a (nontrivial) subspace to see ifthey satisfy the condition. Prove that a subspace is t invariant if and only if itssubbasis has the property that for all of its elements, t( ⃗ β) is in the subspace.̌ 2.37 Is t invariance preserved under intersection? Under union? Complementation?Sums of subspaces?2.38 Give a way to order the Jordan blocks if some of the eigenvalues are complexnumbers. That is, suggest a reasonable ordering for the complex numbers.2.39 Let P j (R) be the vector space over the reals of degree j polynomials. Showthat if j ≤ k then P j (R) is an invariant subspace of P k (R) under the differentiationoperator. In P 7(R), does any of P 0(R), . . . , P 6(R) have an invariant complement?2.40 In P n (R), the vector space (over the reals) of degree n polynomials,andE = {p(x) ∈ P n(R) ∣ ∣ p(−x) = p(x) for all x}O = {p(x) ∈ P n(R) ∣ ∣ p(−x) = −p(x) for all x}are the even and the odd polynomials; p(x) = x 2 is even while p(x) = x 3 is odd.Show that they are subspaces. Are they complementary? Are they invariant underthe differentiation transformation?2.41 Lemma 2.8 says that if M and N are invariant complements then t has arepresentation in the given block form (with respect to the same ending as startingbasis, of course). Does the implication reverse?2.42 A matrix S is the square root of another T if S 2 = T . Show that any nonsingularmatrix has a square root.