HOMEWORK 11 — DUE DEC 15TH

MATH 25

There are two sections: A and B. Please return each part to the proper CA.

A. Problems graded by Benjamin

In the following 4 problems, K is a field which contains Q and M_{n}(K) is the set of all n × n matrices.

A.1. Matrices of trace 0 I. Let M ∈ M_{n}(K) be a matrix whose trace is 0.

(1) Prove that there exists an invertible matrix P such that the first column of P^{−1}MP is

(0 1 0 ⋮ 0) .

(2) Prove that there exists an invertible matrix Q such that all the diagonal entries of Q^{−1}MQ are 0.

A.2. Matrices of trace 0 II. Let

D = (\begin{matrix}1 \\ & 2 \\ & & . & . & . \\ & & & & & n \end{matrix})

and let φ : M_{n}(K) → M_{n}(K) be the map defined by φ(M) = MD − DM.

(1) Prove that ker φ is the set of diagonal matrices. What is the rank of φ?

(2) Prove that im φ is a subspace of the space of matrices whose diagonal entries are 0.

(3) Prove that im φ is the space of matrices whose diagonal entries are 0.

A.3. Matrices of trace 0 III (Shoda’s theorem). Using the previous two problems, prove that a matrix M satisfies Tr(M) = 0 if and only if there exist A, B ∈ M_{n}(K) such that M = AB − BA.

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A.4. Matrices of trace 0 IV. Prove that the subspace of M_{n}(K) generated by all nilpotent matrices is the space of matrices whose trace is 0.

A.5. Skolem-Noether’s theorem. Let φ : M_{n}(K) → M_{n}(K) be a bijective map such that φ(A + λB) = φ(A) + λφ(B), φ(Id) = Id and φ(AB) = φ(A)φ(B). Such a map is called an automorphism of M_{n}(K). The goal of the problem is to find all automorphisms of M_{n}(K).

(1) Check that if P is an invertible matrix, then the map φ_{P} defined by φ_{P} (M) = P MP^{−1} is an automorphism of M_{n}(K).

(2) Let E_{i,j} be the matrix all of whose entries are 0 except the (i, j)-th which is 1. Compute E_{i,j}E_{k,ℓ} (the answer depends on whether j = k or not).

(3) Now let φ be an unspecified automorphism and let U_{i,j} = φ(E_{i,j}). Prove that ker(U_{1,1} − Id) is non-trivial.

(4) Let f_{1} ∈ ker(U_{1,1} − Id) be a non-trivial element and for k ≥ 2, set f_{k} = U_{k,1}(f_{1}). Prove that {f_{1}, ⋯, f_{n}} is a basis of K^{n}.

(5) Let P be the matrix of {f_{i}} in the usual basis of K^{n}. Prove that φ and φ_{P} coincide on the E_{k,1}’s.

(6) prove by recurrence on n that if φ is an automorphism of M_{n}(K), then there exists an invertible matrix Q such that φ(M) = QMQ^{−1}.

B. Problems graded by Inna

B.1. Symmetries. Let f ∈ End(V) be a map such that f^{2} = Id.

(1) Prove that f is diagonalizable, with eigenvalues ±1. Let V_{±} be the eigenspace corresponding to ±1.

(2) Why is such a map called a symmetry (your answer should involve V_{±})?

B.2. Prove that if f ∈ End(V), then V = ker(f) ⊕ im(f) if and only if 0 is a simple root of Π_{f}.

B.3. Projections. Let f ∈ End(V) be a map such that f^{2} = f.

(1) Prove that ker(f) ⊕ im(f) = V.

(2) Prove that f is diagonalizable. What are its eigenvalues, and what are the corresponding eigenspaces?

(3) Why is such a map called a projection?

B.4. Let C_{n}[X] be the space of polynomials with complex coefficients and degree ≤ n. Prove that the map f : C_{n}[X] → C_{n}[X] defined by f(P) = (X^{2} − 1)P′′ is diagonalizable.

HOMEWORK 11 — DUE DEC 15TH 3

B.5. Let A and B be two matrices and let

M = (\begin{matrix}A \\ & B \end{matrix}) .

Find Π_{M} in terms of Π_{A} and Π_{B}.

B.6. Prove that if f : E → E is a diagonalizable map and if V ⊂ E is a subspace stable by f, then the restriction of f to V is diagonalizable. Prove an analogous statement for “triangulable”.

B.7. Let M, N ∈ M_{n}(K) be such that MN − NM = λM where λ ≠ 0.

(1) Prove that M^{k}N − NM^{k} = kλM^{k} for all k ≥ 0.

(2) Prove that M is nilpotent.