AMS 526 Homework 3, Fall 2014
Due: Monday 10/13 in class
1. (15 points) Show that for Gaussian elimination with partial pivoting applied to any matrix A Rnn ,
the growth factor = maxi,j |uij |/ maxi,j |aij | satises 2n1 .
(0)
Solution: Let A = A(0) = aij
AMS526 Homework 2 Solutions
Due : Wednesday 09/23
1. (10 points) Let P Rnn be an orthogonal projection matrix (projector).
(a) Show that I P is also an orthogonal projection matrix.
(b) Show that I 2P is an orthogonal matrix.
Solution
(a) A projector is a
AMS 526 Sample Questions for Test 2
October 22, 2014
Note: The exam is closed-book. However, you can have a single-sided, one-page, letter-size cheat sheet.
1. Answer true or false and give a brief justication. (No credit without justication.)
(a) When so
AMS 526 Homework 5, Fall 2014
Due on Monday 11/17 in class
1. (15 points) The preliminary reduction to tridiagonal form would be of little use if the steps of the QR
algorithm did not preserve this structure. Fortunately, they do.
(a) In the QR factorizat
Sample Solutions for AMS 526 Sample Test 1
September 24, 2014
Note: The exam is closed-book. However, you can have a single-sided, one-page, letter-size cheat sheet.
1. Answer true or false and give a brief justication. (No credit without justication.)
(a
AMS 526 Sample Questions for Final Exam
December 3, 2014
1. (20 points) Answer true or false with a brief justication. (No credit without justication.)
(a) If A Cmm is Hermitian positive denite, then the eigenvalues of BAB T are all real and positive for
AMS 526 Homework 6, Fall 2014
Due: Wednesday 12/3 in class
1. (10 points) In our lectures, it was pointed out that the eigenvalues of a symmetric matrix A Rmm
are the stationary values of the Rayleigh quotient r(x) = xT Ax / xT x for x Rm . Show that the
AMS 526 Homework 4, Fall 2014
Due: Wednesday 10/29 in class
1. (10 points)
(a) Let A Rnm and B = A+ Rmn . Show that the following four relationships hold:
BAB = B
ABA = A
(BA)T = BA
(AB)T = AB
(b) Conversely, show that if A and B satisfy the above four co
AMS526 Homework 2 Solutions
October 8, 2014
1. (10 points) Let P Rnn be an orthogonal projection matrix (projector).
(a) Show that I P is also an orthogonal projection matrix.
(b) Show that I 2P is an orthogonal matrix.
Solution
(a) A projector is a squar
AMS 526 Homework 4 Solutions, Fall 2016
Due: Wednesday 11/02 in class
1. (10 points)
(a) Let A Rnm and B = A+ Rmn . Show that the following four relationships hold:
BAB = B
ABA = A
(BA)T = BA
(AB)T = AB
(b) Conversely, show that if A and B satisfy the abo
AMS 526 Homework 3 Solutions, Fall 2016
Due: Monday 10/19 in class
1. (15 points) Let A = LU be the LU factorization of n-by-n A with |`ij | 1. Let aTi and uTi denote the
ith row of A and U , respectively. Verify the equation
uTi = aTi
i1
X
`ij uTj ,
j=1
AMS 526 Sample Questions for Test 2
October 31, 2016
Note: The exam is closed-book. However, you can have a single-sided, one-page, letter-size cheat sheet.
1. Answer true or false and give a brief justification. (No credit without justification.)
(a) Whe
AMS 526 Homework 1 Sample Solutions
1. (10 points) Show that if A Rmn has rank p, then there exists an X Rmp and Y Rnp
such that A = XY T , where rank(X) = rank(Y ) = p.
Solution:
Let A Rmn be a rank p matrix. This means that there are p linearly independ
AMS 526 Homework 2
Due: Wednesday 09/28 in class
1. (15 points) Suppose that A Rmn has rank n.
(a) (10 points) Show that A(AT A)1 AT is an orthogonal projection matrix onto range of A.
(b) (5 points) Show that kA(AT A)1 AT k2 = 1.
Solution:
(a) Let P = A(
Sample Solutions for AMS 526 Sample Test 1
September 29, 2016
Note: The exam is closed-book. However, you can have a single-sided, one-page, letter-size cheat sheet.
1. Answer true or false and give a brief justification. (No credit without justification.