Math 602
HOMEWORK 1
1. (a) Show that the product of two unitary matrices is also a unitary matrix. (b) If U is unitary, is its inverse U -1 unitary? (c) Is the identity matrix unitary? 2. (a) Show that the determinant of a unitary matrix has absolute valu
Math 602 (2012)
Homework 3
1. Let A be a self-adjoint matrix. Show that there is a self-adjoint matrix B so that B 3 = A(A - 2I)(A - 3I) What are the eigenvalues of B in terms of the eigenvalues of A? What are the eigenvectors of B in terms of the eigenve
Math 602
Homework 4
1. Consider the quadratic form on R3 x2 - 2x2 + x2 + 4x1 x2 + 8x1 x3 + 4x2 x3 = F (x) 1 2 3 Find the following: a) an orthonormal basis of unit vectors cfw_1 , 2 , 3 , and a set cfw_1 , 2 , 3 e e e so that 3 3 2 = x yii F (x) = e i yi
Math 602
Homework 6
1. Show that the pseudoinverse of a vector x Cn is x+ = 0,
1 x
2
if x = 0 x , if x = 0
2. Find the SVD and the pseudoinverse of the following matrices: M= 1 1 -2 -2 L= 10 2 10 2 5 11 5 11
3. Let u1 , . . . , ur be an orthonormal set in
Math 602
Homework 7
The problems here are well known results. You need to show all your work and explanations. 1. (i) Check that the functions 1, sin(nx), cos(nx), (n = 1, 2, 3 . . .) form an orthogonal system in L2 [-, ]. (ii) Normalize them to obtain an
Math 602, 2012
Homework 8
Please note the following very useful inequalities: |f (t)| f and
a
for all t S if f
b b
= sup |f (x)|
xS
f (t) dt
a
f (t) dt
1. Consider the sequence of functions fn (x) = xn . a) Show that the sequence fn is point-wise conver