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LINEAR ALGEBRA BOOT CAMP
WEEK 2: LINEAR OPERATORS
Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero.
The following are facts (in no particular order) that you may assume for the purposes of this worksheet (and, as far as I can tell, for the basic exam).
Theorem 1 (Fund. Thm. of Algebra). Every complex polynomial of degree ≥ 1 has a root.
Definition 2. The characteristic polynomial of an n × n matrix A is the monic polynomial χA(t) ∈ F[t] we obtain by row-reducing tI − A until it is in upper triangular form with monic polynomials p 1 (t),... , pn(t) on the diagonal, and then multiplying the diagonal entries, i.e. χA(t) = p 1 (t) · · · pn(t). All of the eigenvalues of A are roots of χA.
Lemma 3. If A has the block triangular form
A =
[
R ∗
0 S
]
where R and S are square matrices, then χA(t) = χR(t)χS (t).
Fact 4. The determinant det A is the product of the eigenvalues of A (with multiplicity). Equiva- lently, it is (−1)n^ times the constant coefficient of χA. The trace tr A is the sum of the eigenvalues.
Definition 5. If χL(t) = (t − λ)mq(t) with q(λ) 6 = 0 then we say that the algebraic multiplicity of λ is m. The geometric multiplicity of λ is dim ker(L − λI); it is always ≤ m.
Definition 6. Let k be the smallest positive integer such that the set { 1 , L, L^2 ,... , Lk} ⊂ Hom(V, V ) is linearly dependent. Then there exist α 0 ,... , αk− 1 ∈ F such that
Lk^ + αk− 1 Lk−^1 +... + α 1 L + α 0 = 0.
The minimal polynomial of L is defined as
μL(t) = tk^ + αk− 1 tk−^1 +... + α 1 t + α 0.
Lemma 7. Let L : V → V be a linear operator.
(1) If p(L) = 0 for some p ∈ F[t] then μL divides p, i.e. p(t) = μL(t)q(t) for some q(t) ∈ F[t]. (2) λ ∈ F is an eigenvalue of L if and only if μL(λ) = 0.
Lemma 8. A polynomial p(t) ∈ F[t] has λ ∈ F as a multiple root if and only if λ is a root of both p and its derivative p′.
Warmup. This problem should have been on the first worksheet.
(1) Let A and B be two real 5 × 5 matrices such that A^2 = A, B^2 = B, and 1 − (A + B) is invertible. Prove that rank(A) = rank(B). S18-
Eigen–basics. These only require the definition of eigenvector and eigenvalue.
(2) Let V be a complex vector space at T : V → V a linear transformation. Let v 1 ,... , vn be nonzero vectors in V , each an eigenvector of a different eigenvalue. Prove that {v 1 ,... , vn} is linearly independent. F01-10, W02-9, F08-
(3) Must the eigenvectors of a linear transformation T : Cn^ → Cn^ span Cn? Prove it. F08-
(4) Let T : Cn^ → Cn^ be a linear transformation and P (x) a polynomial such that P (T ) = 0. Prove that every eigenvalue of T is a root of P (x). F04-
The Cayley-Hamilton Theorem.
Theorem 9 (Cayley-Hamilton). Let L : V → V be a linear operator on a finite-dimensional vector space. Then L is a root of its own characteristic polynomial; i.e. χL(L) = 0. In particular, the minimal polynomial divides the characteristic polynomial.
The next few exercises outline a proof of this theorem. This is not the shortest proof, nor the most elegant; however, several of the ideas in this outline appear on basic exam questions (probably because this is the proof given in Petersen which is used as the main text for the basic exam).
(5) The cyclic subspace generated by x ∈ V is Cx = span{x, L(x), L^2 (x),.. .}. Clearly Cx is L-invariant. (a) Suppose x 6 = 0. Prove that there is a k ≤ dim(V ) such that x, L(x), · · · , Lk−^1 (x) form a basis for Cx.
(b) What is the matrix representation Cp for L
Cx with respect to the basis from (a)?
(c) What are the characteristic and minimal polynomials of Cp?
(b) Is there an n × n upper triangular matrix A with An^6 = 0 but An+1^ = 0? (Give an example or prove that no such matrix exists). F05-
(13) (a) Find a polynomial P (x) of degree 2 such that P (A) = 0 for ( 1 3 4 2
(b) Prove that P is unique up to multiplication by a constant. S12-
(14) This problem appeared on a Basic Exam in 2014, but it has a mistake. Identify the mistake and provide a counterexample. (a) Find a real matrix A whose minimal polynomial is equal to t^4 + 1.
(b) Show that the usual real linear map v 7 → Av has no nontrivial invariant subspace. S14-
(15) Suppose that V is a finite dimensional vector space over C and that T : V → V is a linear transformation. Suppose that F (x) ∈ C[x] is a polynomial. Show that the linear transformation F (T ) is invertible if and only if F (x) and the minimal polynomial of T have no common factors. S17-
Diagonalizablility. A linear operator T : V → V is diagonalizable if there is a basis of V consist- ing of eigenvectors for T. A matrix A is diagonalizable if there exists an invertible matrix B such that BAB−^1 is diagonal.
(16) Prove that T is diagonalizable if and only if for each eigenvalue λ the geometric multiplicity of λ equals the algebraic multiplicity of λ. An immediate corollary is that if the characteristic polynomial has n distinct roots then T is diagonalizable.
(17) Prove the Minimal Polynomial Characterization of Diagonalizability.
Theorem 10. Let L : V → V be a linear operator on an n-dimensional vector space over F. Then L is diagonalizable if and only if the minimal polynomial factors as
μL(t) = (t − λ 1 ) · · · (t − λk),
where λ 1 ,... , λk are distinct.
(18) Assume A is an n × n complex matrix such that for some positive integer m we have Am^ = In, where In is the n × n identity matrix. Prove that A is diagonalizable. S08-
(19) Let t ∈ R such that t is not an integer multiple of π. Prove that the matrix
A =
cos(t) sin(t) − sin(t) cos(t)
is not diagonalizable. Now do the same for the matrix
B =
1 λ 0 1
F03-
(20) Suppose A is an n × n complex matrix with n distinct eigenvalues. Prove that if B is an n × n complex matrix such that AB = BA then B is diagonalizable. S08-
(21) Suppose that a complex matrix A satisfies
ker((A − λI)) = ker((A − λI)^2 ) for all λ ∈ C.
Show from first principles (i.e. without using the theory of canonical forms) that A must be diagonalizable. F11-
(22) Let V be an n-dimensional vector space (n ≥ 2) over C with basis e 1 ,... , en. Let T be a linear transformation of V satisfying T (e 1 ) = e 2 ,... , T (en− 1 ) = en, T (en) = e 1. (a) Show that T has 1 as an eigenvalue and write down an eigenvector with eigenvalue 1.
(b) Is T diagonalizable? Hint: calculate the characteristic polynomial. F09-
(27) Let A, B be two n×n complex matrices which have the same minimal polynomial M (t) and the same characteristic polynomial P (t) = (t − λ 1 )a^1 · · · (t − λk)ak^ , with distinct λ 1 ,... , λk. Prove that if P (t)/M (t) = (t − λ 1 ) · · · (t − λk) then these matrices are similar. S10-
(28) Let
A =
(a) Find the Jordan form J of A and a matrix P for which P −^1 AP = J.
(b) Compute A^100 and J^100.
(c) Find a formula for an, when an+1 = 4an − 4 an− 1 and a 0 = a, a 1 = b. S10-
(29) Let A be a 3 × 3 matrix with complex entries. Consider the set of such A that satisfy tr(A) = 4, tr(A^2 ) = 6, and tr(A^3 ) = 10. For each similarity (i.e. conjugacy) class of such matrices, give one member in Jordan normal form. The following identity may be helpful:
if
b 1 = a 1 + a 2 + a 3 , b 2 = a^21 + a^22 + a^23 , b 3 = a^31 + a^32 + a^33 ,
then 6 a 1 a 2 a 3 = b^31 + 2b 3 − 3 b 1 b 2.
S11-
(30) Let A be a linear operator on a four dimensional complex vector space that satisfies the polynomial equation P (A) = A^4 + 2A^3 − 2 A − I = 0. Let B = A + I and suppose that the rank(B) = 2. Finally, suppose that | tr(A)| = 2. Give a Jordan canonical form of A. F12-
(31) Suppose A is an endomorphism of a complex vector space with characteristic polynomial
CA(x) = x^4 − 6 x^3 + 13x^2 − 12 x + 4. How many similarity (i.e. conjugacy) classes of elements can have this characteristic poly- nomial? Suppose also that the minimal polynomial MA(x) is equal to CA(x). How many classes satisfy this additional condition? Prove your answers, quoting any general theorems you need. F13-
(32) Let a, b, c, d ∈ R and let
M =
1 0 a b 0 1 0 0 0 c 3 − 2 0 d 2 − 1
(a) Determine the conditions on a, b, c, d so that there is only one Jordan block for each eigenvalue of M in the Jordan form of M.
(b) Find the Jordan form of M when a = c = d = 2 and b = −2. S17-
(33) Consider the matrices
A =
B =
C =
D =
E =
F =
Which pairs of these matrices are similar over R? You must fully justify your answer. S18-
(34) Let T : V → V be a linear operator such that T 6 = 0 and T 5 6 = 0. Suppose that V ∼= R^6. Prove that there is no linear operator S : V → V such that S^2 = T. Does the answer change if V ∼= R^12? F15-
Extra Problems.
(35) Suppose that p ∈ F[t] is a polynomial. Show that deg μp(L)(t) ≤ deg μL(t).
(36) Show that if E : V → V is a projection then tr(E) = dim(im(E)).
(37) Suppose that L is an involution, i.e. L^2 = 1. (a) Show that x ± L(x) is an eigenvector for L with eigenvalue ±1. (b) Show that V = ker(L + I) ⊕ ker(L − I). (c) Conclude that L is diagonalizable.
