# Practice Final Questions with Solutions

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1 Practice Final Questions with Solutions 17th December 2007 Notes on the practice questions The final exam will be on Thursday, Dec. 20, from 9am to 12noon at the Johnson Track, and will most likely consist of 8 12 questions. The practice problems below mostly concentrate on the material from exams 1 2, since you already have practice problems for exam 3. The real final will have plenty of eigenproblems! These questions are intended to give you a flavor of how I want you to be able to think about the material, and the flavor of possible questions I might ask. Obviously, these questions do not exhaust all the material that we covered this term, so you should of course still study your lecture notes and previous exams, and review your homework. Solutions for these practice problems should be posted on the web site by 12/15. List of potential topics: Material from exams 1, 2, and 3, and the problem sets (and lectures) up to that point. Definitely not on final: finite-difference approximations, sparse matrices and iterative methods, non-diagonalizable matrices, generalized eigenvectors, principal components analysis, choosing a basis to convert a linear operator into a matrix, numerical linear algebra and error analysis. Key ideas: The four subspaces of a matrix A and their relationships to one another and the solutions of Ax = b. Gaussian elimination A U R and backsubstitution. Elimination = invertible row operations = multiplying A on the left by an invertible matrix. Multiplying on left by an invertible matrix preserves N(A), and hence we can use elimination to find the nullspace. Also, it thus preserves C(A H ) = N(A). Also, it thus preserves the linear independence of the columns and hence the pivot columns in A are a basis for C(A). Conversely, invertible column operations = multiplying A on the right by an invertible matrix, hence preserving N(A H ) and C(A). Solution of Ax = b when A is not invertible: exactly solvable if b C(A), particular solutions, not unique if N(A) {0}. Least-squares solution A H Aˆx = A H b if A full column rank, equivalence to minimizing Ax b 2, relationship Aˆx = P A b to projection matrix P A = A(A H A) 1 A H onto C(A). The fact that rank(a H A) = rank A = rank A H, hence A H Aˆx = A H b is always solvable for any A, and A H A is always invertible (and positivedefinite) if A has full column rank. Vector spaces and subspaces. Dot products, transposes/adjoints, orthogonal complements. Linear independence, bases, and orthonormal bases. Gram-Schmidt and QR factorization. 1

2 Unitary matrices Q H = Q 1, which preserve dot products x y = (Qx) (Qy) and lengths x = Qx. Determinants of square matrices: properties, invariance under elimination (row swaps flip sign), = product of eigenvalues. The trace of a matrix = sum of eigenvalues. det AB = (det A)(det B), trace AB = trace BA, trace(a + B) = trace A + trace B. For an eigenvector, any complicated matrix or operator acts just like a number λ, and we can do anything we want (inversion, powers, exponentials...) using that number. To act on an arbitrary vector, we expand that vector in the eigenvectors (in the usual case where the eigenvectors form a basis), and then treat each eigenvector individually. Finding eigenvectors and eigenvalues is complicated, though, so we try to infer as much as we can about their properties from the structure of the matrix/operator (Hermitian, Markov, etcetera). SVD as generalization of eigenvectors/eigenvalues. Problem 1 Suppose that A is some real matrix with full column rank, and we do Gram-Schmidt on it to get an the following orthonormal basis for C(A): q 1 = (1, 0, 1, 0, 1) T / 3, q 2 = (1, 2, 1, 0, 0) T / 6, q 3 = ( 2, 1, 0, 0, 2) T / 9. (a) Suppose we form the matrix B whose columns are 3q 1, 6q 2, and 9q 3. Which of the four subspaces, if any, are guaranteed to be the same for A and B? (b) Find a basis for the left nullspace N(A T ). (c) Find a basis for the row space C(A T ). (a) All four subspaces for A and B are the same. Obviously B has the same column space as A, i.e. C(A) = C(B). It follows that they have the same left nullspace, N(A T ) = N(B T ), which are just the orthogonal complement of column space. Since both A and B has full column rank, N(A) = N(B) = {0}. It follows that the row spaces C(A T ) = C(B T ) (b) As we have observed above, N(A T ) = N(B T ), where B = To find a basis for B T, we do Gauss elimination: / / /3, /2 So we may take two vectors (0, 0, 0, 1, 0) T and (5, 2, 1, 0, 6) T as a basis of N(A T ). (c) A is a 5 3 matrix with full column rank, i.e. rank(a) = 3. So the row space have dimension dim(c(a T )) = 3. However, C(A T ) lies in R 3. So C(A T ) = R 3. We may take the vectors (1, 0, 0) T, (0, 1, 0) T, (0, 0, 1) T as a basis. 2

3 Problem 2 Perverse physicist Pat proposes a permutation: Pat permutes the columns of some matrix A by some random sequence of column swaps, resulting in a new matrix B. (a) If you were given B (only), which of the four subspaces of A (if any) could you find? (i.e. which subspaces are preserved by column swaps?) (b) Suppose B = and b = (0, 1, 2) T. Check whether Ax = b is solvable, and if so whether the solution x is unique. (c) Suppose you were given B and it had full column rank. You are also given b. Give an explicit formula (in terms of B and b only) for the minimum possible value of Ax b 2. (a) Only column swaps are performed, so the column space is not changed. As a consequence, the left nullspace is also not changed. So given B, we can find C(A) and N(A T ). (b) Since C(A) = C(B), the solvability of Ax = b is equivalent to the solvability of Bx = b. We do Gauss elimination for the second equation: /3 2/3 So the equation Bx = b, and thus the equation Ax = b, is solvable. Moreover, the solution x is unique. (c) All the vectors Ax are exactly the vectors in the column space C(A) = C(B). Thus the minimum possible value of Ax b 2 is exactly the minimum possible value of Bx b 2. To find the latter one, we need the vector Bx to be the vector in C(B) which is closest to the vector b. In other words, Bx should be the projection of b on C(B). Since B is of full column rank, the projection is Bx = B(B T B) 1 B T b. So the minimum value we want is B(B T B) 1 B T b b 2 Problem 3 Which of the following sets are vector spaces (under ordinary addition and multiplication by real numbers)? (a) Given a vector x R n, the set of all vectors y R n with x y = 3. (b) The set of all functions f(x) whose integral f(x)dx is zero. (c) Given a subspace V R n and an m n matrix A, the set of all vectors Ax for all x V. (d) Given a line L R n and an m n matrix A, the set of all vectors Ax for all x L. (e) The set of n n Markov matrices. (f) The set of eigenvectors with λ < 1 2 of an n n Markov matrix A. (a) This is not a vector space, since it doesn t contain the zero vector. (b) This is a vector space. It is a subset of the vector space consisting all functions. If f and g both has whole integral zero, af + bg also has whole integral zero. So it is a vector subspace. (c) This is a vector space. It is just the column space of A, which is a vector subspace of R m. 3

4 (d) This is not a vector space. A line in R n doesn t have to pass the origin, so the set of all vectors Ax need not contain the zero vector. (e) This is not a vector space. The zero matrix is not a Markov matrix. (f) This is not a vector space in general. If the Markov matrix A has more than one eigenvalue whose absolute value are less than 1/2, then the corresponding set of eigenvectors is not a vector space, since the summation of two eigenvectors corresponding to different eigenvalues is not a eigenvector. Problem 4 The rows of an m n matrix A are linearly independent. (a) Is Ax = b necessarily solvable? (b) If Ax = b is solvable, is the solution necessarily unique? (c) What are N(A H ) and C(A)? (a) Yes. Since the rows of A are linearly independent, rank(a) = m. So the column space of A is an m-dimensional subspace of R m, i.e., is R m itself. It follows that for any b, the equation Ax = b is always solvable. (b) No, the solution maybe not unique. Since rank(a) = m, the nullspace is n m dimensional. Thus the solution is not unique if n > m, and is unique if n = m. (It will never have that n < m, otherwise the rows are not linearly independent.) (c) We have seen in part (a) that C(A) = R m. So N(A T ) = {0}. Since N(A H ) consists those points whose conjugate lies in N(A T ), we see N(A H ) = {0}. Problem 5 Make up your own problem: give an example of a matrix A and a vector b such that the solutions of Ax = b form a line in R 3, b 0, and all the entries of the matrix A are nonzero. Find all solutions x. Such a matrix must be an m 3 matrix whose nullspace is one dimensional. In other words, the rank is 3 1 = 2. We may take A to be an 2 3 matrix whose rows are linearly independent. As an example, we take ( ) ( A =, b = ) To find all solutions, we do elimination ( ) ( ) so the solutions are given by x = (1 t, t, 0) T. ( ) ,

5 Problem 6 Clever Chris the chemist concocts a conundrum: in ordinary least-squares, you minimize Ax b 2 by solving A H Aˆx = A H b, but suppose instead that we wanted to minimize (Ax b) H C(Ax b) for some Hermitian positive-definite matrix C? (a) Suppose C = B H B for some matrix B. Which of the following (if any) must be properties of B: (i) Hermitian, (ii) Markov, (iii) unitary, (iv) full column rank, (v) full row rank? (b) In terms of B, A, and b, write down an explicit formula for the x that minimizes (Ax b) H C(Ax b). (c) Rewrite your answer from (b) purely in terms of C, A, and b. (d) Suppose that C was only positive semi-definite. Is there still a minimum value of (Ax b) H C(Ax b)? Still a unique solution x? (a) For C = B H B a Hermitian positive-definite matrix, B must be (iv) full column rank. To show this, we only need to notice that x Cx = x B H Bx = Bx 2. Since C is positive definite, B has no nonzero nullspace. It follows that B is of full column rank. B don t have to be Hermitian or Markov or unitary or full row rank. For example, we may 1 0 ( ) take B = Then C = B H B = (b) We have (Ax b) H C(Ax b) = (Ax b) H B H B(Ax b) = (BAx Bb) H (BAx Bb) = BAx Bb 2. To minimize this, we need to solve the equation (BA) H (BA)ˆx = (BA) H Bb. Since B and A are of full column rank, BA is of full column rank. So (BA) H (BA) is invertible. The solution is given by ˆx = (A H B H BA) 1 A H B H V b. (c) Since B H B = C, we have ˆx = (A H CA) 1 A H Cb. (d) If C is only positive semi-definite, we still have C = B H B for some B, but B need not to be of full column rank. As above, we have (Ax b) H C(Ax b) = BAx Bb 2. So the minimum value of (Ax b) H C(Ax b) still exists. However, since B may be not of full column rank, the matrix need not be invertible. The solution x need not be unique. For example, if we take C to be the zero matrix, any x will minimize (Ax b) H C(Ax b). Problem 7 True or false (explain why if true, give a counter-example if false). (a) For n n real-symmetric matrices A and B, AB and BA always have the same eigenvalues. [Hint: what is (AB) T?] (b) For n n matrices A and B with B invertible, AB and BA always have the same eigenvalues. [Hint: you can write det(ab λi) = det((a λb 1 )B). Alternative hint: think about similar matrices.] (c) Two diagonalizable matrices A and B with the same eigenvalues and eigenvectors must be the same matrix. 5

6 (d) Two diagonalizable matrices A and B with the same eigenvalues must be the same matrix. (e) For n n matrices A and B with B invertible, AB and BA always have the same eigenvectors. (a) True. Since A and B are real-symmetric matrices, (AB) T = B T A T = BA. But (AB) T has the same eigenvalue as AB. So AB and BA have the same eigenvalues. (b) True. Since B is invertible, we have AB = B 1 (BA)B. So BA is similar to AB, and they must have the same eigenvalues. (c) True. Let Λ be the diagonal matrix consists of eigenvalues, and S be the matrix of eigenvectors (ordered as the eigenvalue matrix). Then we have A = S 1 ΛS and also B = S 1 ΛS. So A = B. (d) False. For example, they are different. ( ) 1 0 and 0 2 ( ) 1 2 are both diagonalizable, with the same eigenvalues, but 0 2 (e) False. ( ) ( ) ( ) ( ) For example, let A = and B =. Then AB = and BA = The eigenvectors of AB are (0, 1) T and (1, 0) T, while the eigenvectors or BA are (0, 1) T and (1, 1) T. Problem 8 You are given the matrix A = (a) What is the sum of the eigenvalues of A? (b) What is the product of the eigenvalues of A? (c) What can you say, without computing them, about the eigenvalues of AA T? (a) The sum of the eigenvalues of A equals trace of A. So the sum of the eigenvalues of A is = 1. (b) The product of the eigenvalues of A equals the determinant of A. We do row transforms / /2 0, / where in the first step we add the other rows to the first row. eigenvalues equals det(a) = 2. It follows that the product of (c) AA T is a real symmetric matrix, so its eigenvalues are all real and nonnegative. Since A is nonsingular, all eigenvalues of AA T are positive. The product of all eigenvalues of AA T are 6

7 det(aa T ) = det(a) det(a T ) = det(a) 2 = 4. The sum of all eigenvalues of AA T is trace(aa T ) = i,j a2 ij = 9. Problem 9 You are given the quadratic polynomial f(x, y, z): f(x, y, z) = 2x 2 2xy 4xz + y 2 + 2yz + 3z 2 2x + 2z. (a) Write f(x, y, z) in the form f(x, y, z) = x T Ax b T x where x = (x, y, z) T, A is a realsymmetric matrix, and b is some constant vector. (b) Find the point (x, y, z) where f(x, y, z) is at an extremum. (c) Is this point a minimum, maximum, or a saddle point of some kind? (a) We have A = 1 1 1, b = (b) We compute the partial derivatives to find the extremum point: f = 4x 2y 4z 2 = 0 x f = 2x + 2y + 2z = 0 y f = 4x + 2y + 6z + 2 = 0 x The equation is just 2Ax = b. The solution to the the equations above is x = 1, y = 1, z = 0. So the extreme point is (1, 1, 0). (c) We look for the pivots of A: / Since A is positive definite, the extreme point is a minimum. Problem 10 Suppose A is some diagonalizable matrix. Consider the vector y(t) = e A2t x for some vector x. (a) If A is 3 3 with eigenvalues λ 1, λ 2, and λ 3 with eigenvectors x 1, x 2, and x 3, and x = x 1 + 3x 2 + 4x 3, what is y(t)? (b) If lim t y(t) = 0 for every vector x, what does that tell you about the eigenvalues of A and of e A2? (c) In terms of A, give a system of differential equations that y(t) satisfies, and the initial condition. 7

8 (d) In terms of A, give a linear recurrence relation that y(t) satisfies for t = k t for integers k and some fixed t. (a) A 2 has eigenvalues λ 2 1, λ 2 2 and λ 2 3 with eigenvectors x 1, x 2 and x 3, so e A2t has eigenvalues e λ2 1 t, e λ2 2 t and e λ2 3 t with the same eigenvectors. Thus y(t) = e A2t x = e A2t (x 1 + 3x 2 + 4x 3 ) = e λ2 1 t x 1 + 3e λ2 2 t x 2 + 4e λ2 3 t x 3. (b) If lim t y(t) = 0 for every vector x, we must have Re λ 2 1 < 0, Re λ 2 2 < 0 and Re λ 2 3 < 0. The eigenvalues e λ2 1, e λ 2 2 and e λ 2 3 all sastisfies e λ 2 < 1. (c) y(t) satisfies the system y = A 2 y. The initial condition is y(0) = x. (d) Denote by y k = y(k t). Then equation y(t) = e A2t x gives the recurrence relation y k+1 = e A2 t y k with initial condition y 0 = x. Problem 11 Suppose A is an m n matrix with full row rank. Which of the following equations always have a solution (possibly non-unique) for any b? (a) Ax = b (b) A H x = b (c) A H Ax = b (d) AA H x = b (e) A H Ax = A H b (f) AA H x = Ab (a) The equation Ax = b has a solution for any b, as explained in problem 4. (b) The equation A H x = b may has no solution, since the column space of A H is an m- dimensional subspace in the whole space R n. (c) The equation A H Ax = b may has no solution, since the C(A H A) = C(A H ) is an m- dimensional subspace in R n. (That C(A H A) = C(A H ) comes from the fact N(A H A) = N(A) which we proved in class.) (d) The equation AA H x = b will always has a unique solution, since AA H is an m m matrix whose rank is m, i.e. AA H is invertible. (e) The equation A H Ax = A H b will always has a solution. In fact, any solution to Ax = b (from part (a)) is always a solution to A H Ax = A H b. (f) The equation AA H x = Ab will always has a unique solution, since AA H is invertible, as explained in part (d). Additional Practice Problems Be sure to look at: (i) Exams 1, 2, and 3. (ii) The practice problems for exam 3. stuff.) (iii) Ideally, also review your homework problems. (The above problems mostly cover non-eigenvalue 8

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