Numerical Linear Algebra (MTH 451)
Sample Problems for Exam 1
1. Consider the 3 ×3 matrix
A=
120
210
001
Compute ||A||1,||A||∞,||A||F.
2. For the matrix A=3 1
1 3
(a) Compute the SVD of A.
(b) Plot the right and left singular vectors.
(c) Show that the singular values of A are the absolute values of the
eigenvalues of A.
(d) What is ||A||2?
(e) What is the 2-norm condition number of A?
(f) What is the rank of A?
3. Do any two of the following:
(a) Let || · || be any norm on Cnas well as the induced matrix norm
on Cn×n. Let A be an n×nmatrix. Show that
ρ(A)≤ ||A||
where ρ(A) = max{|λ|;λan eigenvalue of A}.
(b) Let u∈Cmand v∈Cn. Prove that ||uv∗||2=||u||2||v||2.
(c) Let Abe an n×nmatrix. Define ||A||max = max1≤i,j≤n|aij |. Show
that ||A||max is a matrix norm.
1