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 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
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
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
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 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: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 1: Course Overview;
Matrix Multiplication
Yan Yu
Stony Brook University
Yan Yu
Numerical Analysis I
1 / 22
Outline
1
Course Overview
2
Matrix Notation and
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 7: LU Factorization;
Gaussian Elimination with Pivoting
Yan Yu
Stony Brook University
Yan Yu
Numerical Analysis I
1 / 27
Outline
1
Gaussian Elimination(MC3
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 2: Algorithmic Consideration;
Orthogonality; Vector Norms
Yan Yu
Stony Brook University
Yan Yu
Numerical Analysis I
1 / 20
Outline
1
Algorithmic Considerat
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.
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 6: Accuracy and Stability;
Triangular Systems;
Backward Stability of Back Substitution
Yan Yu
Stony Brook University
Yan Yu
Numerical Analysis I
1 / 21
Out
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 3: Matrix Norms;
Singular Value Decomposition
Yan Yu
SUNY Stony Brook
Yan Yu
Numerical Analysis I
1 / 14
Outline
1
Matrix Norms (NLA 3)
2
Singular Value De
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 5: Conditioning and Condition Number;
Floating Point Arithmetic
Yan Yu
SUNY Stony Brook
Yan Yu
Numerical Analysis I
1 / 15
Outline
1
Conditioning and Condi
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 4: More on SVD; Projectors
Yan Yu
SUNY Stony Brook
Yan Yu
Numerical Analysis I
1 / 18
Outline
1
Singular Value Decomposition (NLA4-5)
2
Projectors (NLA6)
Y
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 2: Algorithms and Efficiency;
Block Matrices and Algorithms;
Range and Null Space
Xiangmin Jiao
Stony Brook University
Xiangmin Jiao
Numerical Analysis I
1 / 34
Outline
1
Algorithms and Effic
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 8: Accuracy and Stability of
Gaussian Elimination
Yan Yu
Stony Brook University
Yan Yu
Numerical Analysis I
1 / 16
Outline
1
Condition Number of Gaussian E
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 4 Solutions, Fall 2015
Due: Wednesday 10/28 in class
1. (10 points)
(a) Let A 2 Rnm and B = A+ 2 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
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
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
AMS 526 Sample Questions for Test 1
September 28, 2017
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) Wheth
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
AMS526: Numerical Analysis I
(Numerical Linear Algebra for
Computational and Data Sciences)
Lecture 9: Positive-Denite Systems;
Cholesky Factorization
Yan Yu
Stony Brook University
Yan Yu
Numerical Analysis I
1 / 12
Outline
1
Positive-Denite Systems (MC4.
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 5: More on Projectors;
Conditioning and Condition Number
Xiangmin Jiao
SUNY Stony Brook
Xiangmin Jiao
Numerical Analysis I
1 / 19
Outline
1
More on Projectors (NLA6)
2
Conditioning and Condit
AMS526: Numerical Analysis I
(Numerical Linear Algebra)
Lecture 8: Accuracy and Stability of
Gaussian Elimination
Xiangmin Jiao
Stony Brook University
Xiangmin Jiao
Numerical Analysis I
1 / 22
Outline
1
Condition Number of Gaussian Elimination (NLA22)
Per
AMS 526 Homework 2
Due: Thursday 09/28 in class
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.
2. (10 points) Consider the mat