Math 2371, Final Exam, Spring 2009
Instructor: D. Swigon
1
NAME: _
Show all your work, justify your results.
Problem 1: (20 points)
(a) Show that if is a complex diagonal matrix, then there is a complex diagonal unitary
matrix U such that = U .
(b) Show t
Math 2371, Midterm Exam practice, Spring 2009 Instructor: D. Swigon SOLUTIONS
1
Problem 1: Let A be an anti-selfadjoint map on a finite-dimensional complex Euclidean space. Show that (a) A - I is invertible. (b) If U = ( A + I )( A - I ) -1 then U is unit
Math 2371, Final Exam, Spring 2009 Instructor: D. Swigon NAME: _ Show all your work, justify your results.
1
Problem 1: (20 points) (a) Show that if is a complex diagonal matrix, then there is a complex diagonal unitary matrix U such that = U . (b) Show t
Math 2371 Spring 2007 Quiz #4 Solution
2 -9 2 Problem 1: Show that the eigenvalues of A = 10 1 - 5 have positive real parts 1 4 3 without computing any eigenvalues.
3 4 1 1 2 - 1 and determinants of the principal minors of B Proof: We have B = A + A* = 3
Math 2371 Spring 2009 Quiz #5 SOLVE ONE OF THE PROBLEMS Problem 1: Prove or disprove: If a square complex matrix has a unique polar decomposition then it is invertible. Proof: The statement is true. Let us prove the equivalent statement: If A is singular
Math 2371 Spring 2009 Quiz #1 Solutions Problem 1: Suppose that a map T on a complex Euclidean space is idempotent, i.e., T 2 = T . Show that if T is normal then it is self-adjoint. Proof: Let x be an eigenvector of T and the corresponding eigenvalue. The
Math 2371 Spring 2007 Quiz #2 Solutions Problem: Let N be a normal map. Show that N = max | j | where j are the
j
eigenvalues of N. Proof: By definition, N = max Nx = max ( Nx, Nx) . Suppose x = j x j , where x j are
x =1 x =1
mutually orthogonal eigenvec
Math 2371 Spring 2009 Quiz #7
I Problem 1: Let | . | be an induced matrix norm on C nn . (a) Show that | I | = 1 . (b) Show that if A in C nn is such that | A | < 1 then I
1 1 | ( I A) 1 | 1+ | A | 1 | A |
Solution: (a) By definition, | I | = max
x0
Ix x
Math 2371 Spring 2009 Quiz #3 Problem 1: (a) Show that in a complex Euclidean space, if A is such that ( x, Ax) > 0 for all x 0 , then A is self-adjoint. (b) Show that the statement is not true in a real Euclidean space.
Proof: (a) Suppose that a map A is
Math 2371 Spring 2009 Quiz #6
R Problem 1: Let X = I n be a normed space with norm | x | p and X be its dual space.
Show that if X is represented as I n then the dual norm on X is the q-norm, i.e., R | l | = | l | q , where 1 / p + 1 / q = 1 .
Solution: E
Math 2371, Midterm Exam practice, Spring 2009 Instructor: D. Swigon Problem 1: Let A be an anti-selfadjoint map on a finite-dimensional complex Euclidean space. Show that (a) A I is invertible. (b) If U = ( A + I )( A I ) 1 then U is unitary and U I is in
Math 2370, Final Exam, Fall 2008 Instructor: D. Swigon NAME: _ Show all your work, justify your results. Number and sign every page. Problem 1 (15 points): Suppose that x and y are vectors in a linear space V and M V is a subspace. Let K be the space span
1
Math 2371, Midterm Exam practice, Spring 2009
Instructor: D. Swigon
SOLUTIONS
Problem 1: Let A be an anti-selfadjoint map on a finite-dimensional complex Euclidean space.
Show that
(a) A I is invertible.
(b) If U = ( A + I )( A I ) 1 then U is unitary a
Linear Algebra Preliminary Exam Topics
a) Geometry of vector spaces: Subspaces, linear independence, bases, dimension, isomorphism,
linear functionals, dual space
(see, e.g., Lax 1, 2; HJ 0)
b) Geometry of linear operators: Range, kernel, rank, projection
Math 2371 Spring 2009
Quiz #1 Solutions
Problem 1: Suppose that a map T on a complex Euclidean space is idempotent, i.e., T 2 = T .
Show that if T is normal then it is self-adjoint.
Proof: Let x be an eigenvector of T and the corresponding eigenvalue. The
Math 2371 Spring 2007
Quiz #2 Solutions
Problem: Let N be a normal map. Show that N = max | j | where j are the
j
eigenvalues of N.
Proof: By definition, N = max Nx = max ( Nx, Nx) . Suppose x = j x j , where x j are
x =1
x =1
mutually orthogonal eigenvec
Math 2371 Spring 2009
Quiz #3
Problem 1:
(a) Show that in a complex Euclidean space, if A is such that ( x, Ax) > 0 for all x 0 , then A
is self-adjoint.
(b) Show that the statement is not true in a real Euclidean space.
Proof: (a) Suppose that a map A is
Math 2371 Spring 2007
Quiz #4
Solution
2 9 2
Problem 1: Show that the eigenvalues of A = 10 1 5 have positive real parts
1
4
3
without computing any eigenvalues.
3
4 1
1 2 1 and determinants of the principal minors of B
Proof: We have B = A + A* =
3 1
Math 2371 Spring 2009
Quiz #5
SOLVE ONE OF THE PROBLEMS
Problem 1: Prove or disprove:
If a square complex matrix has a unique polar decomposition then it is invertible.
Proof: The statement is true. Let us prove the equivalent statement: If A is singular
Math 2371 Spring 2009
Quiz #6
R
Problem 1: Let X = I n be a normed space with norm | x | p and X be its dual space.
Show that if X is represented as I n then the dual norm on X is the q-norm, i.e.,
R
| l | = | l | q , where 1 / p + 1 / q = 1 .
Solution: E
Math 2371 Spring 2009
Quiz #7
I
Problem 1: Let | . | be an induced matrix norm on C nn .
(a) Show that | I | = 1 .
(b) Show that if A in C nn is such that | A | < 1 then
I
1
1
| ( I A) 1 |
1+ | A |
1 | A |
Solution:
(a) By definition, | I | = max
x0
Ix
Math 2371, Midterm Exam practice, Spring 2009
Instructor: D. Swigon
Problem 1: Let A be an anti-selfadjoint map on a finite-dimensional complex Euclidean space.
Show that
(a) A I is invertible.
(b) If U = ( A + I )( A I ) 1 then U is unitary and U I is in