MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS Instructor: Shmuel Friedland Department of Mathematics, Statistics and Computer Science email: friedlan@uic.edu Last update April 18, 2010 1 HOMEWORK ASSIGNMENT 1 Assigned 1-13-10 Due 1-22-10 Do the following problems 1. Let u = (1, 1, 1, 1), v = (2, 0, 2, 1). Find (a) The cosine of the angle between u and v. (b) The scalar and the vector projection of v on u. (c) A basis to the orthogonal complement of U := span(u, v). (d) The projection of the vector (1, 1, 0, 0) on U and U. 2. Let A R 4 3. Assume that the vector (1, 1, 1, 1) is a vector in the column space of A. Is it possible that a vector (2, 0, 2, 1) is in the null space of A? If yes give an example of such a matrix. If not, justify why. 3. Consider the overdetermined system (a) Is this system solvable? x 1 + x 2 + x 3 = 4 x 1 + x 2 + x 3 = 0 x 2 + x 3 = 1 x 1 + x 3 = 2 (b) Find the least squares solution of this system. (c) Find the projection of (4, 0, 1, 2) on the column space of the coefficient matrix A R 4 3 of this system. 4. Let ( 1, 0), (0, 1), (1, 3), (2, 9) be four points in the plane (x, y) Find (a) The best least squares fit by a linear function y = ax + b. (b) The best least squares by a quadratic polynomial y = ax 2 + bx + c. (c) Explain briefly why there exist a unique cubic polynomial y = ax 3 +bx 2 + cx + d passing through these four points. 1
5. Let a t 1 < t 2 <... < t n b be n points in the interval [a, b]. For any two continuous functions f, g C[a, b] define f, g := n i=1 f(t i)g(t i ). Let P m be the vector space of all polynomials of degree at most m 1. (a) Show that for m n, is an inner product on P m. (b) Is, an inner product on P n+1? Justify! 6. For the inner product f, g := 1 1 f(x)g(x)dx on C[ 1, 1] Find the cosine of the angle between f(x) = 1 and g(x) = e x. 2 HOMEWORK ASSIGNMENT 2 Assigned 1-26-10 Due 2-3-10 Do the following problems from Schaum s Outline of Linear ALgebra by S. Lipschutz and M. Lipson, 4th edition, pages 258-261: 7.58, 7.60, 7.64, 7.71, 7.75, 7.77 (Use the Gram-Schmidt algo in problem 7.77 to find the QR decomposition of A = [v 1, v 2, v 3 ]), 7.91, 7.92, 7.94, 7.95. 3 HOMEWORK ASSIGNMENT 3 Assigned 2-1-10 Due 2-10-10 [1]: 2.2 page 14: Problems 2,3; 2.3 page 19: Problem 12. [3]: 6.4 p 363-365, Problems: 4(a,b,c,e); 5(a,b,f),6,10,12,14. 4 HOMEWORK ASSIGNMENT 4 Assigned 2-9-10 Due 2-17-10 [3]: 1.4 p 77, Problems: 7(a,d).. 7.2 p 434, Problem 2a. Do as follows: Consider the augmented matrix  := [A b] and doe the Gauss elimination as in LU decomposition of A. Obtain the matrix B = [U c]. Write down the corresponding system of equations for U x = c. Solve it by backward substitution. So first solve the third equation for x 3. After that solve the second equation for x 2. Then the first equation for x 1. 6.7 pages 403-404, Problems 1(a,d), 4(a), 5(d), 8. 6.6 page 396, Problems 7 (c,f), 10. 5 HOMEWORK ASSIGNMENT 5 Assigned 2-16-10 Due 2-24-10 1. Let A H n, i.e. A = [a pq ] n p=q=1 Cn n is n n hermitian matrix. Let D k = det A k, where A k = [a pq ] k p,q=1 for k = 1,..., n. (So D 1,..., D n are the leading principle minors of A.) 2
(a) Show that det A is a real number. (b) Show that D 1,..., D n are real. (c) Show that A = LDL, where L is a lower triangular matrix with 1 s on the main diagonal, and D = diag(d 1,..., d n ) is a diagonal matrix where d 1,..., d n are all nonzero real numbers if and only if D 1,..., D n are nonzero. (This is the analog of LDL decomposition for real symmetric matrices.) (d) Assume that D 1,..., D n are nonzero. Show that d i = D i D i 1 for i = 1,..., n, where D 0 = 1. (e) Let the assumption of 1d. Show that the number of positive and negative eigenvalues of A is the number of positive and negative numbers in the sequence d 1,..., d n. (Hint: Use Sylvester s law of inertia: If A, B H n and B = T AT for some invertible matrix then the inertia of A and B are the same. (See Math 320 notes, p 277 for definition of inertia of symmetric matrices, which is the same definition for the inertia of Hermitian matrices.) (f) Show that positive definite: A 0, i.e. x Ax > 0 for any 0 x C n, if and only if D k > 0 for k = 1,..., n. (g) Show that negative definite: A 0, i.e. x Ax < 0 for any 0 x C n, if and only if ( 1) k D k > 0 for k = 1,..., n. 2. Find the inertia of the following real symmetric and hermitian matrices without finding the eigenvalues. State if the matrix is positive definite, negative definite or indefinite. (A H n is indefinite if A has a positive and a negative eigenvalue.) (Hint: Use the results of 1e - 1g.) (a) [ 2 6 6 17 ] [ ; (b) 5 3 + 4i 3 4i 6 ] ; (c) 1 i 2 + i i 2 1 i 2 i 1 + i 2 3. Let A H n, i.e. A = [a pq ] n p=q=1 Cn n is n n hermitian matrix. Let λ 1 (A)... λ n (A) be the n eigenvalues of A, where each eigenvalue repeats according to its multiplicity, i.e. the characteristic polynomial of A is det(zi A) = (z λ 1 (A))... (z λ n (A). Show (a) For each integer k [1, n] one has the inequality λ 1 (A) a kk λ n (A). (Hint: Use the Rayleigh quotient estimate for x = e k = (δ 1k,..., δ nk ). (b) λ 1 (A) max(a 11,..., a nn ), λ n (A) min(a 11,..., a nn ). (c) Give two examples of 3 3 real symmetric matrix A with diagonal entries a 11 > a 22 > a 33, where in the first one λ 2 (A) < a 33 and the second one where λ 2 (A) > a 11. (Hint: Try rank one symmetric matrices.) (d) Show that for each integer k [2, n 1] k λ j (A) j=1 k a jj j=1 k λ n j+1 (A). (Hint: Use Ky-Fan inequalities, i.e.theorem 2.42 on page 23 of my Math 425 notes. j=1. 3
(e) Let α 1 [ α 2 be the two ] eigenvalues of the 2 2 principle submatrix of a11 a A: A 2 := 22. Show that a 21 a 22 λ 1 (A) α(a), λ 2 (A) β, β λ n (A), α λ n 1 (A). (Hint: Use Theorem 2.36 on page 21 of Math 425 notes for A and A.) (f) Let M be the maximum value of a ij, i, j = 1,..., n. Show that nm λ 1 (A) and λ n (A) nm. (Hint: Estimate the Rayleigh quotient for A from below and above accordingly.) Give examples of matrices where each of the bounds is best possible. (Hint: Use a correct rank one matrix.) (g) Use the results of 3e and 3f to estimate from below and above the eigenvalues of the matrix given in Problem 2 (c). 6 HOMEWORK ASSIGNMENT 6 Assigned 3-1-10 Due 3-12-10 1. Problems [3, p 380-382], 1-6,8. 2. Problems [1, p 36], 3,4,9 (In 9 you can assume that A is a normal matrix.) 3. Problems [3, p 453-455], 1(a,e), 4, 10, 11, 15, 19. 7 HOMEWORK ASSIGNMENT 7 Assigned 3-17-10 Due 4-2-10 For A R m n let A R n m be the Moore-Penrose inverse. 1. Let A R m n where n > m and A = [B 0], where B R m m and is invertible. (a) Show [ A B 1 = 0 ], (A ) = [(B ) 1 0]. (b) Find the Moore-Penrose inverse for the matrices [3, p 380] problem 2(c,d). (c) Give the least squares solution of the system Ax = b with the minimal norm x, where A are given in 1b, where all the coordinates of b are equal to 1. 2. Let A R 4 3 be given as in [3, p 381] problem 5. Find A. Give the least solution of the system Ax = b with the minimal norm x, where all the coordinates of b are equal to 1. 3. Problems [1, p 44-45], 1,2,3. 4. Problems [3, p 351-352], 1(c,d), 7, 8(a,b), 9. 4
8 HOMEWORK ASSIGNMENT 8 Assigned 3-31-10 Due 4-9-10 1. [1]: Problems 1,2a,3 (Recall that A nonderogatory if its minimal polynomial is equal to the characteristic polynomial. The Jordan block J k (λ) is defined on p 48, definition 3.12). 2. [4]: Problems 9.61-9.65, 9.67. 9 HOMEWORK ASSIGNMENT 9 Assigned 4-19-10 Due 4-26-10 Problem 1. Assume that A, B C n n are two nilpotent matrices of index of nilpotence l, m respectively. Assume that AB = BA, i.e. A and B commute. 1. Show that AB and A + B are nilpotent. 2. Show that the index of nilpotency of AB and A + B is at most l + m. Problem 2. 1. Prove that the matrix and Jordan form. 1 1 1 1 1 1 1 1 0 is nilpotent, and find its invariants 2. Prove that the matrix in part 1. is not similar to the matrix Problem 3. 1 1 1 1 1 1 1 0 0 1. Let J k (λ) C k k be the Jordan block defined in M425 lecture notes, p 48, Definition 3.12, [1]. Show that J k (λ) is similar to J k (λ). 2. Show that any A C n n is similar to A. Problem 4. Find all possible Jordan canonical forms for the matrices whose characterisitc polynomial is p(z) and the minimal polynomial ψ(z) as follows: 1. p(z) = (z 2) 4 (z 3) 2, ψ(z) = (z 2) 2 (z 3) 2, 2. p(z) = (z 7) 7, ψ(z) = (z 7) 3. Problem 5. Let A R n n and assume that all eigenvalues of A are real. Show that there exists an invertible matrix T R n n such that T 1 AT is the Jordan canonical form of A. Problem 6. Do Problem 1 on page 69 of [1].. 5
References [1] S. Friedland, Linear Algebra II, Lectures Notes, Spring 2010, http://www2.math.uic.edu/ friedlan/lectnotesm425s10.pdf [2] G.H. Golub and C.F. Van Loan. Matrix Computation, John Hopkins Univ. Press, 3rd Ed., Baltimore, 1996. [3] S.J. Leon, Linear Algebra with Applications, Prentice Hall, 6th Edition, 2002. [4] S. Lipschutz and M. Lipson, Linear Algebra, Fourth Edition, Schaum s Outlines, McGraw-Hill, 2009. 6