ESE 415 Optimization Assignment 1
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ESE 415 Optimization
Assignment 1
2022
1. Eigenvalues and eigenvectors of a square matrix play important roles in optimization, for example, in the determination of convergence of gradient methods. For each of the following n × n matrices A ∈ Mn ×n (F),
(a) Determine all the eigenvalues of A.
(b) For each eigenvalue λ of A, find the set of eigenvectors corresponding to λ .
(c) If possible, find a basis for F consisting of eigenvectors of A.
(d) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q−1AQ = D .
(i) A = [3(1) 2(2)] for F = R
(ii) A = 「(l)
1 
2 
1(3)
for F = R
(iii) A = [2(i) 
i] for F = C, where i = ^− 1
(iv) A = 「(l)
for F = R
2. Given the matrix A = l 0(1) 1(1)
「 0 0
2(0) 」
3 
,
(i) The eigenvalues of A are 1, 1, 3. How many linearly independent eigenvectors does A have?
(ii) Denote λ 1 = 3 and λ2 = 1 and the respective eigenvectors v1 and v2 . If v3 ∈ R3
satisfies (A − λ2 I)v3 = v2 , what is the value of k such that (A − λ2 I)k v3 = 0.
3. Let A be an n × n matrix and let λ(A) = {λ1 , . . . ,λn } be the set of eigenvalues of A. Show the following propositions described in the note.
(a) λ(cI + A) = {c + λ1 , . . . ,c + λn }, where I is the identity matrix. (b) λ(Ak ) = {λ1(k) , . . . ,λn(k)}.
(c) If A is nonsingular, then λ(A−1) = { 
, . . . , 
}.
(d) λ(A) = λ(AT ).
4. Sketch the following sets, and state if each set is a linear subspace and/or linear variety.
(a) S1 = {x ∈ R3 ' x1 = x2 , x2 = x3 }
(b) S2 = {x ∈ R3 ' x1 + x2 + x3 = 1 }
5. Consider the matrix A, given by
A = l
45(.)2 
0 80(.)」
「8.0 − 1.0 1.0 
.
For Gaussian Elimination A is represented in terms of its LU factors, i.e., A = PLU . Compute the matrices P , L, and U .
6. Consider the vector-valued function,
f(x) = f(x1 ,x2 ) = l
o22sx2」
「 4x1(3) + 7x1 x2(2) 
.
What is the Jacobian of this function at the point x = [ ]2(1) ?
7. (a) Consider the following system of linear equations,
x1 + 2x2 + 4x3 = 1
x1 + 3x2 + 9x3 = 6
x1 + 5x2 + 25x3 = 4.
Is the solution of this system unique? If yes, why? If not, how many solutations are there?
(b) Repeat the questions in (a) for the following system,
x1 + 2x2 + 4x3 = 1
x1 + 3x2 + 9x3 = 6
2x1 + 6x2 + 18x3 = 12.
8. (Bonus Problem) Show that the collection of all polynomial functions of the form f(x) = an xn + an −1xn −1 + ... + a1 x + a0 ,
defined on the interval [a,b] with real coefficients, ai ∈ R, forms a real vector space.
2022-09-06