MATH 456 Homework 1
Due 1/29/2023 by 11:59pm (upload to Canvas)
Submission Instructions:
Your submission should contain two total files. One pdf file containing all of your solutions (including
the output of your code) and one file with your code. I should not need to run your code to view
your solutions.
1. (10 pts) Prove that a square matrix and its transpose have the same characteristic polynomial,
and therefore the same set of eigenvalues.
2. (15 pts) An eigenvalue and eigenvector of the matrix A may be evaluated by solving the system
of nonlinear equations
(A− λI)x = 0
xTx = 1
for the unknowns λ and x. Using Newton’s method, starting with estimates λ0 and x0, show
that the next iteration is determined by
A∆x−∆λx0 − λ0∆x = −(A− λ0I)x0
−xT0 ∆x =
1
2
(xT0 x0 − 1)
where ∆x = x1 − x0 and ∆λ = λ1 − λ0. Comment on the difference between this method and
the method of inverse iteration.
Hint: For a vector valued function f(z) with z = Rn, Newton’s method has the form
zn+1 = zn − Jf(zn)−1f(zn)
where Jf(zn) is the Jacobian of f evaluated at zn. Note, you will not need to compute the
inverse to derive the above expression.
3. (10 pts) Assume that A is a 3× 3 matrix with the given eigenvalues. Decide to which eigenvalue
Power Iteration will converge and determine the convergence rate constant S.
(a) {1, 2, 7} (b) {8,−9, 10}
4. (10 pts) Ca
y out two steps of inverse iteration for the matrix
A =
(
2 2
2 5
)
using the eigenvalue estimate λ̃ = 5 and the initial vector v0 = (1, 1)
T . Verify that the elements of
the vector v2 agree wth those of the true eigenvector with an accuracy of about 5%. Evaluate the
Rayleigh quotient using the vector v2, and verify that the result agrees with the true eigenvalue
to about 1 in 3000.
1
Computer Problems (You may use Matlab or Python)
Submit your code and a table of iterates containing your results.
5. (15 pts) Let
A =
ï£«ï£ 5 2 −2−12 −19 12
−12 −22 15

(a) Apply 10 steps of the Power Method with initial vectors v = (1, 1, 1)T to estimate the
dominant eigenvalue of A.
(b) Apply 10 steps of the Inverse Power Method with shift 0 and initial vector v = (1, 1, 1)T
to estimate the eigenvalue closest to zero.
Provide a table of iterates for each method’s approximation to the eigenvalue.
6. (20 pts) Consider the symmetric matrix
A =
ï£«ï£ XXXXXXXXXX
1 1 4
 .
With initial guess v0 = (
1√
3
, 1√
3
, 1√
3
)T , apply the following methods:
(a) Rayleigh quotient iteration
(b) Inverse power iteration with a shift of 5.
Provide a table of iterates for each method’s approximation to the eigenvalue and comment on
the speed at which each method converges.
2