Problems in Linear Algebra and Representation Theory

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1 Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific order. In some cases a problem may depend on one of the previous problems on the list. No effort has been made to keep the list organized. Notation and terminology. Unless explicitly indicated otherwise, all matrices under consideration have entries in an arbitrary field (in particular, of arbitrary characteristic), and I denotes an identity matrix of the appropriate size. The symbol F q denotes a finite field with q elements (it is unique up to isomorphism), and whenever the notation F q is used, it is implicitly assumed that q is a power of a prime number. If n N and k is a field, we write M n (k) for the ring of n n matrices over k; it is a (unital, associative) k-algebra, which is noncommutative unless n = 1. In most cases the linear algebra terminology used in the problems below agrees with that introduced in S. Axler s Linear Algebra Done Right. This book is also recommended for most of the background needed to understand and solve these problems. (1) Let A be a 2 2 matrix such that A 10 = 0. What can you say about A 5? (2) Let V be an inner product space over R or C, and let e 1, e 2,..., e n be a collection 1 of vectors in V, not necessarily orthonormal. Let, denote the inner product in V, and form the matrix n-by-n matrix A with entries e i, e j (where i, j = 1,..., n). Prove or disprove the following statement: the vectors e i are linearly independent if and only if the matrix A is nonsingular. (3) Let K be a field, and let F be a subfield of K. Assume that F is infinite. Let p(x) be a polynomial in one variable with coefficients in K, and suppose that p(a) F whenever a F. Show that the coefficients of p lie in F. [Hint: learn about the Vandermonde determinants if you don t already know about them.] Give an example to show that the statement is false without the assumption that F is infinite. (4) Let A be the n-by-n matrix over a field of characteristic 0, all of whose entries are 1. List the eigenvalues of A, counted with their multiplicities. Is A diagonalizable? (5) Show that if A is a square complex matrix with A k = I (where I is the identity matrix of the same size as A) for some positive integer k, then A is diagonalizable. (6) Let T be a linear operator on a finite dimensional complex vector space, and p(x) a polynomial in 1 variable with complex coefficients. Prove or disprove: λ C is an eigenvalue of T if and only if p(λ) is an eigenvalue of p(t ). What about the following statement: if µ C is an eigenvalue of p(t ), then there exists an eigenvalue λ of T such that p(λ) = µ? 1 Here, n has nothing to do with the dimension of V. 1

2 2 (7) Consider a finite dimensional complex inner product space. Show that every normal operator on this space has a square root. Give an example of a (non-normal) operator that DOES NOT have a square root. (8) Compute the number of k-dimensional subspaces of an n-dimensional vector space over F q (where 0 k n < ). (9) Compute the orders of the groups GL n (F q ) and SL n (F q ), where n N. (10) Compute the number of matrices A M n (F q ) for which A 2 = 0, as a function of n and q. (11) Describe the conjugacy classes in the groups GL 2 (F q ) and SL 2 (F q ). (The shape of the answer may depend on q.) In particular, compute their number. (12) Let A be a finite dimensional associative unital algebra over a field k. Prove that A is an integral domain if and only if A is a division ring. (13) Describe all left ideals, all right ideals and all two-sided ideals in the ring M n (C) of n n matrices over C. (14) Prove that any C-algebra automorphism of M n (C) has the form X AXA 1 for some fixed A GL n (C). (15) Show that every finite dimensional division algebra 2 over R is isomorphic to either R, or C, or the quaternion algebra H. (16) Let f : M n (C) M n (C) be a C-linear map (not necessarily an algebra homomorphism). Prove that there exist matrices A 1,..., A d M n (C) and B 1,..., B d M n (C) such that d f(x) = A j XB j X M n (C). j=1 (17) Fix n N. Show that the set of unordered n-tuples of points of C can be naturally identified with C n. Show that the set of unordered n-tuples of points of CP 1 can be identified with CP n. (Here, CP n is the n-dimensional projective space over C.) (18) Let A M n (R). Show that det(e A ) = e tr A. Here, det denotes the determinant, tr denotes the trace, and the exponential, e A exp(a), of a matrix A is defined by the usual power series e A := I + A A A3 + (19) Which matrices B M 2 (R) can be written as B = e A for some A M 2 (R)? (20) Draw a picture of the set of nilpotent 2 2 matrices over R. (21) Is the group SL n (C) connected? What about SL n (R)? 2 A division algebra is an associative unital (but not necessarily commutative) algebra in which every nonzero element is invertible.

3 (22) Recall that SO(3, R) denotes the special orthogonal group of real 3 3 matrices, i.e., the group of those A M 3 (R) for which A A t = I (where A t is the transpose of A) and det A = 1. (a) What is the dimension of SO(3, R) as a manifold? (b) Show that SO(3, R) is connected. (c) Show that any element g SO(3, R) is a rotation about an axis in R 3. (d) Compute the fundamental group of SO(3, R). (23) The next problem has to do with describing the universal cover of SO(3, R). Let H denote the quaternion algebra over R, with basis 1, i, j, k. [Recall that the multiplication in H is determined by the identities i 2 = j 2 = k 2 = 1 and ij = k = ji.] Given u = x + yi + zj + wk H, where x, y, z, w R, we write u = x yi zj wk. Write V = Ri Rj Rk H, and U = { u H uu = 1. (a) Show that U is a group under multiplication. (b) Show that if u U and v V, then uvu 1 V. (c) Thus we obtain a map T : U GL(V ), u T u, where T u (v) = uvu 1. Show that T is a group homomorphism. (d) Describe the kernel and the image of T. (24) Let n 2, and let A be the (unital and associative, but noncommutative) C-algebra with generators x 1,..., x n and relations x i x j + x j x i = 2δ ij. (a) Find the dimension of A over C. (b) Construct a simple A-module. (At least, find the dimension of a simple A-module.) [Hint: part (b) is fairly difficult. You should try the cases n = 2, 3, 4 before approaching the problem for general n.] (25) Given A = (a ij ) M n (C), recall that A denotes the conjugate transpose of A, i.e., A = (a ji ). We say that A is positive, and write A 0, if A = A and all eigenvalues of A (which are automatically real in view of the first condition) are nonnegative. More generally, if A, B M n (C), we write A B if and only if B A 0. This defines a partial order (but not a total order!) on M n (C). Learn the definition of extremal points of subsets of real and complex vector spaces, and find the extremal points of the following subsets of M n (C): (a) { X M n (C) 0 X I ; (b) { X M n (C) XX I. (26) If S is an ellipsoid in R n centered at the origin, then, in suitable orthonormal coordinates x 1,..., x n on R n, S can be defined by the inequality n j=1 a jx 2 j 1, where 0 < a 1 a 2 a n are uniquely determined by S. Let H denote a hyperplane in R n passing through the origin; then S H is an ellipsoid in H, and H is a Euclidean space of dimension n 1. Hence S H determines a sequence of coefficients 0 < b 1 b 2 b n 1. Prove that a 1 b 1 a 2 b 2 a n 1 b n 1 a n. 3

4 4 (27) Let S be an ellipsoid in R n (as in the previous problem), and let P be a parallelepiped such that S is inscribed in P (in particular, each face of P is tangent to S). However, we do not require the edges of P to be parallel to the axes of S; in particular, S does not determine P uniquely. Show, nevertheless, that the length of the main diagonal of P depends only on S. Can you find a formula for this length in terms of the coefficients a 1,..., a n introduced in the previous problem? (28) Let V be a vector space over R or C. A subset V 0 V is called a cone if it is nonempty and for all v 1, v 2 V 0 and all λ R with λ 0, we have λv 1 + v 2 V 0. Given a cone V 0 V, we define a subset (V ) 0 V as follows: φ (V ) 0 v V 0, φ(v) R and φ(v) 0. Check that (V ) 0 is a cone in V. Now let V = M n (C) and V 0 = { A M n (C) A 0. (a) Check that V 0 is a cone in V. (b) Describe (V ) 0. (c) Show that ( (V ) ) 0 = V 0 under the natural identification V (V ). (29) Let X = U(n) = { B GL n (C) BB = I be the group of n-by-n unitary matrices over C, viewed as a real manifold. Fix a diagonal n-by-n matrix A with real entries, whose diagonal entries, call them a 1,..., a n, are pairwise distinct. [Of course, usually, A U(n).] Define a (smooth) function f : U(n) R, B Re ( tr(ab) ), where Re(z) is the real part of a complex number z. (a) Find the critical points of f. (b) (This is harder.) Find the second differential of f at each of these points. 3 (30) Fix n N, and let h n denote the Heisenberg Lie algebra of dimension 2n + 1 (over any given field k). Namely, h n is the Lie algebra with basis x 1,..., x n, y 1,..., y n, c and with the Lie bracket defined by [x i, y j ] = δ ij c, [x i, x j ] = [y i, y j ] = [x i, c] = [y j, c] = 0 (where 1 i, j n and δ ij is the Kronecker delta). Find the maximal possible dimension of an abelian Lie subalgebra of h n. (31) Prove that any SO(3, R)-invariant real-valued polynomial function on R 3 has the form f(x, y, z) = p(x 2 + y 2 + z 2 ) for some p R[t]. 3 Recall that if X is a smooth manifold and f : X R is a smooth function, then, in general, there is no natural way to define the second differential, or Hessian, of f at a given point x X. However, if x is a critical point of f, i.e., the first differential d x f : T x X R vanishes, then d 2 xf can be canonically defined; it is a symmetric bilinear form on the tangent space T x X. For details, see, for example, John Milnor s book Morse Theory.

5 (32) Prove that any GL n (C)-conjugation invariant C-valued polynomial function on M n (C) has the form f(a) = F (p 0 (A),..., p n 1 (A)) for some F C[x 0, x 1,..., x n 1 ], where the functions p j are defined by the identity p 0 (A) + tp 1 (A) + + t n 1 p n 1 (A) + t n = det(ti A). (33) Show that for each c C, the set Σ c = { A M n (C) tr(a) = c, rk(a) = 1 is a single GL n (C)-conjugacy class. (a) Find dim Σ c (the dimension of Σ c as an algebraic variety over C). (b) Draw a picture of Σ c for n = 2. (c) For which c C is the set Σ c closed? (d) For which c C is the closure Σ c of Σ c a smooth submanifold of M n (C)? (34) Let A, B M n (C) be such that rk(ab BA) 1. Show that there exists a basis of C n with respect to which A and B are simultaneously upper-triangular. (35) Let k be a field of characteristic 0, and let V be a finite dimensional vector space over k. Consider the space of symmetric n-tensors, S n V = (V V V ) Sn, where = k and the symmetric group S n acts on the n-fold tensor product V n = V V V by permuting the factors. Show that S n V is spanned by elements of the form v v, v V. n (36) In the situation of problem (35), let n V V V denote the subspace of skew-symmetric tensors. (a) Is it true that V V = S 2 V 2 V? (b) Is it true that V V V = S 3 V 3 V? (c) Can one remove the assumption that char k = 0 in either (a) or (b)? (37) Let A be an associative algebra with a unit over a field k. Fix n > 1. Define a k-algebra structure on the vector space A n = A k k A by n (a 1 a n ) (a 1 a n) := (a 1 a 1) (a n a n). (a) Show that S n A and n A are closed under multiplication in A n provided A is commutative. (b) Can one drop the commutativity assumption? (c) Assume that char k = 0 and A is commutative. Show that S n A is generated, as an algebra, by elements of the form a 1 +1 a a, a A. n 1 n 2 n 1 5

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