18.06 Professor Edelman Quiz 3 December 5, 2011
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1 18.06 Professor Edelman Quiz 3 December 5, 2011 Grading 1 Your PRINTED name is: Please circle your recitation: 1 T Kestutis Cesnavicius kestutis 2 T Niels Moeller moller 3 T Kestutis Cesnavicius kestutis 4 T Niels Moeller moller 5 T Yan Zhang yanzhang 6 T Taedong Yun tedyun
2 1 (24 pts.) Let A = (4 pts) True or False: The matrix A is Markov. True. Markov matrices have columns that sum to 1 and have nonnegative entries. The answer of false applies to what is known as Positive Markov Matrices. 2. (6 pts) Find a vector x 0 and a scalar λ such that A T x = λx. The obvious choice is (1,1,1) with λ =1 as this is the column sum property. Also easy to see is (1,0,0) with λ =
3 3. (4 pts) True or False: The matrix A is diagonalizable. (Explain briey.) True. The three eigenvalues, on the diagonal, are distinct. 4. (4 pts) True or False: One singular value of A is σ = 0. (Explain briey.) False. The matrix is nonsingular, since it has no zero eigenvalues. Nonsingular square matrices have all n singular values positive. 5. (6 pts) Find the three diagonal entries of e At as functions of t. They are e t, e 0.5t, e 0.9t. 3
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5 2 (30 pts.) 1. (5 pts) An orthogonal matrix Q satises Q T Q = QQ T = I. What are the n singular values of Q? They are all 1. The singular values are the positive square roots of the eigenvalues of QQ T = Q T Q = I (10 pts) Let A = 2. Find an SVD, meaning A = UΣV T, where U 3 σ 1 and V are orthogonal, and Σ = σ 2 is diagonal with σ 1 σ 2 σ 3 0. (Be sure that the factorization is correct and satisies all stated requirements.) A = The singular values are in decreasing order and are positive. One can compute AA T and A T A, but easier to rig the permutation matrices and correct the sign. σ 3 5
6 3. (15 pts) The 2 2 matrix A = σ 1 u 1 v1 T + σ 2 u 2 v2 T, where σ 1 > σ 2 > 0 and both u 1, u 2 and v 1, v 2 are orthonormal bases for R 2. The set of all vectors x with x = 1 describes a circle in the plane. What shape best describes the set of all vectors Ax with x = 1? Draw a general picture of that set labeling all the relevant quantities σ 1, σ 2, u 1, u 2 and v 1, v 2. (Hint: Why are u 1, u 2 relevant and v 1, v 2 not relevant?) The svd rotates (or reects) the circle with V T, scales to an ellipse with axes in the coordinate directions through Σ, and then a rotated ellipse with axes in the direction u 1 and u 2 after U is applied. The Σ scales the x and y axes by σ 1 and σ 2 respectively, and σ 1 is the longer of the two. 6
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8 3 (16 pts.) 1. (6 pts) Let x 0 be a vector in R 3. How many eigenvalues of A = xx T are positive? zero? negative? (Explain your answer.) (Hint: What is the rank?) A is symmetric pos semidenite and rank 1, so there are 1 positive, 2 zero, and no negative eigenvalues. 2. (6 pts) a) What are the possible eigenvalues of a projection matrix? 0 and 1 (Since P 2 = P, λ 2 = λ.) b) True or False: every projection matrix is diagonalizable. True, every projection matrix is symmetric, hence diagonalizable. 3. (4 pts) True or False: If every eigenvalue of A is 0, then A is similar to the zero matrix. False. A Jordan block with zero eigenvalues is not similar to the zero matrix for n > 1. 8
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10 4 (30 pts.) Consider the matrix A = x with parameter x in the (1,1) position. 1. (10 pts) Specify all numbers x, if any, for which A is positive denite. (Explain briey.) No x, the matrix is clearly singular with two equal rows and two equal columns. 2. (10 pts) Specify all numbers x, if any, for which e A is positive denite. (Explain briey.) The eigenvalues of e A are the exponentials of the eigenvalues of the matrix A. Since A is symmetric the eigenvalues are real, and thus exponentials are positive. A symmetric matrix with positive eigenvalues is positive denite. 10
11 3. (10 pts) Find an x, if any, for which 4I A is positive denite. (Explain briey.) One can take any x < 3. The easiest choice is x = 1. With this guess the matrix has two eigenvalues 0 and one eigenvalue 3 both less than 4, so 4 λ > 0 for all three eigenvalues. Systematically, one can consider the three upper left determinants of 4I A which are 4 x, 11 3x, and 24 8x. They are all positive if and only if x < 3. 11
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Lecture 11: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 11 1 / 11 Definition The n n matrix A is diagonalizableif there exits nonsingular matrix P d 1 0 0. such that P 1 AP = D, where
The matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0.
) Find all solutions of the linear system. Express the answer in vector form. x + 2x + x + x 5 = 2 2x 2 + 2x + 2x + x 5 = 8 x + 2x + x + 9x 5 = 2 2 Solution: Reduce the augmented matrix [ 2 2 2 8 ] to