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1 Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and normal if A A = AA : Find the dimension and a basis for each of the following vector spaces. (a) M n (C) ; n n complex matrices, over C: (b) M n (C) over R (c) H n (C) ; n n Hermitian matrices, over R (d) H n (R) ; n n real symmetric matrices, over R (e) S n (C) ; n n skew-hermitian matrices, over R (f) The space consisting of all real polynomials of A over R; where A = ! 0 0 0! ;! = + p 3i 2 Is H n (C) a vector space over C? IS the set of n n normal matrices a subspace of M n (C)? Show that M n (C) = H n (C) + S n (C) ; i.e., any n n matrix is a sum of Hermitian matrix and a skew-hermitian matrix. 2. Find the space of matrices commuting with (a) A = I n (b) A = 0 a 0 (c) A = ; a 6= b 0 b (d) A = B A (e) All n n matrices 3. True or False. If true, what is the dimension? Basis? (a) (x; y) : x 2 + y 2 = 0; x; y 2 R is a subspace of R 2 : (b) (x; y) : x 2 + y 2 = 0; x; y 2 C is a subspace of C 2 : (c) (x; y) : x 2 y 2 = 0; x; y 2 R is a subspace of R 2 : (d) f(x; y) : x y = 0; x; y 2 Rg is a subspace of R 2 : (e) f(x; y) : x y = ; x; y 2 Rg is a subspace of R 2 : (f) fp (x) : p (x) 2 P [x] has degree 3g is a subspace of P [x] : (g) fp (x) : p (0) = 0; p (x) 2 P [x]g is a subspace of P [x] (h) fp (x) : 2p (0) = p ()g is a subspace of P [x] : (i) fp (x) : p (x) 0; p (x) 2 P [x]g is a subspace of P [x] : 4. Show that M 2 (R) = W W 2 ; where a b W = : a; b 2 R b a

2 c W 2 = d d c : c; d 2 R : 5. Let A be an n n real matrix. (a) Show that if A t = A and n is odd, then jaj = 0: (b) Show that if A 2 + I = 0; then n must be even. (c) Does (b) remain true for complex matrices? 6. Introduce the correspondence between complex numbers and real matrices: x y z = x + iy~z = 2 M y x 2 (R) ; and de ne for each pair of complex numbers u and v : q = (u; v) u v = C (q) = 2 M v u 2 (C) (a) Show that z~z t : (b) Show that ZW = W Z (c) Show that z~z and w~w imply zw + ZW: (d) Find Z n ; where z = r (cos + i sin ) : (e) What is the matrix corresponding to i? (f) Show that jc (q)j 0: Find C (q) when juj 2 + jvj 2 = : (g) Replace each entry of C (q) with the corresponding 2 2 real matrix to the entry to get U V R (q) = V t U t 2 M 4 (:) Then jr (q)j 0: (h) Show that R (q) is similar to a matrix of form U X X U for some X: (i) Show that R (q) is singular if and only if C (q) is singular if and only if u = v = 0: 7. True or false (a) For any m n matrix A with rank r; there exists invertible m m and n n matricesp and Q such that Ir 0 A = P Q: 0 0 (b) For any n n matrix A with rank r; there exists an invertible n n matrix P such that Ir 0 A = P P 0 0 : (c) If A is a real matrix and A exists over C; then A is also a real matrix. (d) If (A ) 2 = A 2 ; then A = A or A = A: (e) If rank A = rank B; then rank A 2 = rank B 2 (f) rank (A + B) rank (A) + rank (B) 2

3 (g) rank (A B) rank (A) rank (B) (h) Since (; i) and (i; ) are linearly independent over R; the matrix i is invertible. i 8. Suppose that A and B are both m n matrices. Show that Ax = 0 and Bx = 0 have the same solution space if and only if there exists an invertible matrix C such that A = CB: Use this fact to show that if rank A 2 = rank (A) ; then there exists an invertible matrix D such that A 2 = DA: 9. What are the matrices that are similar to themselves only? 0. Prove assertions (a) and (b) and construct an example for (c). (a) Let A 2 M n (C) : If the eigenvalues of A are distinct from each other, then A is diagonalizable, i.e., there is an invertible matrix P such that P AP is diagonal. (b) If matrix A commutes with a matrix with distinct eigenvalues, then A is diagonalizable. (c) Give an example of a matrix A that is diagonalizable but not unitary diagonalizable, that is, P AP is diagonal for some invertible P; but U AU is not diagonal for any unitary matrix U:. True or false (a) If A k = 0 for all positive integers k 2; then A = 0: (b) If A k = 0 for some integer k; then tr A = 0: (c) If tr A = 0; then jaj = 0: (d) If A and B are similar, then jaj = jbj (e) If A and B are similar, then they have the same eigenvalues. (f) If A and B have the same eigenvalues, then they are similar. (g) If A and B have the same characteristic polynomial, then they have the same eigenvalues. (h) If A and B have the same eigenvalues, then they have the same characteristic polynomial. (i) If A and B have the same characteristic polynomial, then they are similar. (j) If tr A k = tr B k for all positive integers k; then A = B: (k) If the eigenvalues of A are ; :::; n ; then A is similar to the diagonal matrix diag f ; :::; n g : (l) diag f; 2; :::; ng is similar to diag fn; :::; 2; g (m) If A has a repeated eigenvalue, then A is not diagonalizable. (n) If A is diagonalizable, then A is normal. (o) If A is unitarily diagonalizable, then A is normal. (p) If A has r nonzero eigenvalues, then rank (A) r: 2. Let A 2 M n (C) and A 6= 0: De ne a transformation on M n (C) by T (X) = AX XA; X 2 M n (C) Show that 3

4 (a) T is linear (b) Zero is an eigenvalue of T (c) If A k = 0; then T 2k = 0: (d) If A is diagonalizable, so is T. (e) T (XY ) = XT (Y ) + T (X) Y: (f) If A and B commute, so do T and L where L is de ned as L (X) = BX XB; X 2 M n (C) : Find all A such that T =0 and discuss the converse of (f). 3. Let W be an invariant subspace of a linear transformation A on a nitedimensional vector space V: (a) Show that if A is invertible, then W is also invariant under A : (b) If V = W W 0 ; is W 0 necessarily invariant under A? 4. Show that if A is an invertible Hermitian matrix, then there exists and invertible matrix P such that P AP = A : 5. Is it possible for some non-hermitian matrix A 2 M n (C) to satisfy x Ax 0 for all x 2 R n? x 2 C n? 6. Construct examples (a) Matrices A and B that have only positive eigenvalues, AB has only negative eigenvalues. (Note that A and B are not necessarily Hermitian). (b) Is it possible that A + B has only negative eigenvalues for matrices A and B with positive eigenvalues? (c) Matrices A; B, and C are positive de nite, ABC has only negative entries. (d) Is it possible that the matrices in (c) are 3 3? 7. Let A 2 M n (C) be a normal matrix. Show that (a) ker A = ker A (b) Im A = Im A (c) C n = Im A ker A 8. A permutation matrix is a matrix which has exactly one in each row and each column. (a) How many n n permutation matrices are there? (b) The product of two permutation matrices of the same size is also a permutation matrix. How about the sum? (c) Show that any permutation matrix is invertible and its inverse is equal to its transpose. (d) For what permutation matrices P; does P 2 = I? 9. Let P be the n n permutation matrix 4

5 In..... = B A Then show the following (a) For any positive integer k n; 0 P k In = k ; and I k 0 P n = P t ; P n = I n (b) P; P 2 ; :::; P n are linearly independent. (c) P i + P j is a normal matrix (d) For n 3; P is diagonalizable over C; but not over R: (e) For every P i ; there exists a permutation matrix T such that T P i T = P: 20. Let A be a linear transformation on an inner product space V: Show that for any unit vector x 2 V (Ax; x) (x; Ax) (Ax; Ax) : In particular, for A 2 M n (C) and x 2 C n with jjxjj = ; x A xx Ax x A Ax: 2. Let e ; e 2 ; :::; e n be vectors of an inner product space over a eld F; and let A = (a ij ) ; where a ij = (e i ; e j ) ; i; j = ; 2; :::; n: Show that e ; e 2 ; :::; e n are linearly independent if and only if A is nonsingular. 22. Let V be an inner product space over R: (a) If e ; e 2 ; e 3 are three vectors in V with pairwise product negative, that is, (e i ; e j ) < 0; i; j = ; 2; 3; i 6= j; show that e ; e 2 ; e 3 are linearly independent. (b) Is it possible for three vectors in the xy-plane to have pairwise negative products? (c) Does (a) remain valid when the word "negative" is replaced with "positive"? (d) Suppose that u; v; and w are three unit vectors in the xy-plane. What are the maximum and minimum values that (u; v) + (v; w) + (w; u) can attain? and when? 23. If fe ; :::; e n g is an orthonormal basis for an inner product space V over C; and x 2 V; show that x = P n i= (x; e i) e i 5

6 and (x; x) P k i= j(x; e i)j 2 ; k n: Why are pairwise orthogonal nonzero vectors linearly independent? 24. Let W be a subspace of an inner product space V and let S be a subset of V: Answer true or false. (a) There is a unique subspace W 0 such that W 0 + W = V: (b) There is a unique subspace W 0 such that W 0 W = V: (c) There is a unique subspace W 0 such that W 0 W = V and (w; w 0 ) = 0 for all w 2 W and w 0 2 W 0 = (d) W?? = W: (e) S?? = S? (f) S?? = S? (g) S? + W (h)?? S? \ W = = S?? \ W? S?? + W? 6

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