Linear Algebra, 2020a

422 Chapter Five. Similarity
̌ 3.31 Find the eigenvalues and associated eigenvectors of the differentiation operator
d/dx: P 3 → P 3 .
3.32 Prove that the eigenvalues of a triangular matrix (upper or lower triangular)
are the entries on the diagonal.
̌ 3.33 This matrix has distinct eigenvalues.
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1 2 1
6 −1 0
−1 −2 −1
(a) Diagonalize it.
(b) Find a basis with respect to which this matrix has that diagonal representation.
(c) Draw the diagram. Find the matrices P and P −1 to effect the change of basis.
̌ 3.34 Find the formula for the characteristic polynomial of a 2×2 matrix.
3.35 Prove that the characteristic polynomial of a transformation is well-defined.
3.36 Prove or disprove: if all the eigenvalues of a matrix are 0 then it must be the
zero matrix.
̌ 3.37 (a) Show that any non-⃗0 vector in any nontrivial vector space can be an
eigenvector. That is, given a ⃗v ≠ ⃗0 from a nontrivial V, show that there is a
transformation t: V → V having a scalar eigenvalue λ ∈ R such that ⃗v ∈ V λ .
(b) What if we are given a scalar λ? Can any non-⃗0 member of any nontrivial
vector space be an eigenvector associated with λ?
̌ 3.38 Suppose that t: V → V and T = Rep B,B (t). Prove that the eigenvectors of T
associated with λ are the non-⃗0 vectors in the kernel of the map represented (with
respect to the same bases) by T − λI.
3.39 Prove that if a,..., d are all integers and a + b = c + d then
( ) a b
c d
has integral eigenvalues, namely a + b and a − c.
̌ 3.40 Prove that if T is nonsingular and has eigenvalues λ 1 ,...,λ n then T −1 has
eigenvalues 1/λ 1 ,...,1/λ n . Is the converse true?
̌ 3.41 Suppose that T is n×n and c, d are scalars.
(a) Prove that if T has the eigenvalue λ with an associated eigenvector ⃗v then ⃗v is
an eigenvector of cT + dI associated with eigenvalue cλ + d.
(b) Prove that if T is diagonalizable then so is cT + dI.
̌ 3.42 Show that λ is an eigenvalue of T if and only if the map represented by T − λI
is not an isomorphism.
3.43 [Strang 80]
(a) Show that if λ is an eigenvalue of A then λ k is an eigenvalue of A k .
(b) What is wrong with this proof generalizing that? “If λ is an eigenvalue of A
and μ is an eigenvalue for B, then λμ is an eigenvalue for AB, for, if A⃗x = λ⃗x and
B⃗x = μ⃗x then AB⃗x = Aμ⃗x = μA⃗x = μλ⃗x”?
3.44 Do matrix equivalent matrices have the same eigenvalues?
3.45 Show that a square matrix with real entries and an odd number of rows has at
least one real eigenvalue.
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