Math 1080: Spring 2010
Homework #10 (due April 15)
Problem 1:
Let Q and R be the QR factors of a symmetric tridiagonal matrix H. Show that the product
K = RQ is again a symmetric tridiagonal matrix.
(Hint: Prove the symmetry of K. Show that Q has Hessenbe
Math 1080: Spring 2005
Homework #3
Problem 1:
Show that the Householder reflector H = I 2 ww T , with w = 1 , is symmetric and
orthogonal. Find the eigenvalues and eigenvectors of H.
SOLUTION:
Symmetry:
(
H T = I 2 ww T
Orthogonality:
)T = I 2(ww T )T = I
Math 1080: Spring 2011
Homework #4
Problem 1:
Determine the relative condition number for the following mathematical problems:
a) f ( x ) = cos( x )
b) f ( x ) = x =
c) f ( x ) =
i=1 x i2
n
1
1+ x 2
SOLUTION:
Bu definition, for a differentiable scalar fu
Math 1080: Spring 2005
Homework #2
SOLUTIONS
Problem 1:
Let A be an m n matrix ( m n ), and let A = QR be a reduced QR factorization. Show
that A has full rank (i.e., rank n) if and only if all the diagonal entries of R are nonzero.
SOLUTION:
If A has ful
Math 1080: Spring 2011
Homework #1
SOLUTIONS
Problem 1:
Consider the matrix
1 / 2 0 3 / 2
Q= 0
1
0
3 / 2 0 1 / 2
Show that Q is orthogonal. What transformation of IR 3 does it correspond to ?
(Hint: Find the vector a that is invariant under Q. Pick a v
MATH 1080: Spring 2011
Final Exam Review Topics
Reviews for both Midterm I and Midterm II
PLUS
Chapters: V.24-29
Theory:
Definition and properties of eigenvalues and eigenvectors of a matrix
Definition and properties of characteristic polynomial
Definitio
MATH 1080: Spring 2011
Midterm Exam I Review Topics
Chapters: I.1-2, II.6-8, 10-11
Theory:
Definition and properties of scalar & outer products
Definition and properties of an orthogonal matrix
Definition and properties of the Euclidean norm (Cauchy Schwa
MATH 1080: Spring 2011
Midterm Exam II Review Topics
Chapters: III.12-16, IV.20-23
Theory:
Understanding the distinction between conditioning, stability and accuracy
Definition of absolute and relative condition numbers
Basic axioms of floating-point arit
1
Math 1080, Midterm Exam II, Spring 2005
Instructor: D. Swigon
SOLUTIONS
Problem 1: a) (10 points) Determine the relative condition number for the following problem.
Are there values of x for which the problem is ill-conditioned? Justify your answer.
1 e
Math 1080, Midterm Exam, Spring 2011
Instructor: D. Swigon
SOLUTIONS
Problem 1: (25 points)
a) Find the orthogonal projector P onto range(A) where
1 0
A = 0 2
1 1
b) Show that Q = I 2P is an orthogonal matrix.
SOLUTION:
a) The projector is given by P =
Math 1080: Spring 2011
Homework #11 (due April 22)
Problem 1:
a) Determine the matrices U, , V in the singular value decomposition, A = UV T , of
the following matrices
1 1 1 1
A 1 = 1 1 1 1
1 1 1
0
A2
2 2
2 1
=
0
1
2 3
0 2
1 3
1 1
1 1
b) Use the re
Math 1080: Spring 2011
Homework #9 (due April 8)
Problem 1:
Calculate the Rayleigh quotients r1 = r ( x 1 ) and r 2 = r ( x 2 ) for the following matrix A and vectors x1 and
x2. How close are r1 and r2 to an eigenvalue of A ?
1 4 1
1.5
1
4 6 4
2
A=
x1 =
Math 1080: Spring 2011
Homework #8 (due April 1)
Problem 1:
Find the diagonalization A = XX 1 of the following matrix:
1
0
A=
0
0
0 4
2
0
3
0
1 0
3
0
3 1
SOLUTION:
In order to calculate the decomposition we must first determine the eigenvalues and eigen
Math 1080: Spring 2011
Homework #7 (due March 16)
Problem 1:
a
wT
Show that if A = 11
is symmetric and positive definite, then a 11 > 0 and both
w K
T
ww
K and K
are symmetric and positive definite.
a 11
(Hint: Use the fact that A is positive definite
Math 1080: Spring 2011
Homework #6
SOLUTIONS
Problem 1:
Let A and B be two nonsingular lower triangular m m matrices. Show that the product AB is
also lower triangular.
SOLUTION :
The matrix A is lower triangular if and only if a ij = 0 whenever i < j . S