Description

Ding Zhao
24-677 Special Topics: Linear Control Systems
• At the beginning of each question you will find the key words for the knowledge that the exercise will help you to practice. • Right after each week’s recitation, we have an about half-hour homework Q&A session. I request you to work on the assignment early and bring your questions to this session to take advantage of this support.
• Note that we do not have programming questions in Homework 4.
Exercise 1. eigenvalues (practice solving the eigenvalues manually instead of using Python, the same for the entire homework 4)
For A
Find the eigenvalues of A1 and A2, such that the eigenvalues are in ascending order
(λ1 ≤ λ2 ≤ λ3)
Exercise 2. Singular values.
For A
1. Find the eigenvalues of A .
2. Find the eigenvalues of A .
3. Find the singular values of A1
4. Find the singular values of A2
Write eigenvalues in ascending order.
Exercise 3. Characteristic polynomial
A
A matrix in this form is called a companion form matrix
1. Calculate the characteristic polynomial in terms of αi,i = 1,2,3,4.
2. Derive the normalized eigenvector v of A in terms of its eigenvalues λ, s.t. kvk2 = 1.
(As the solution is analytical, we will check it manually. No AutoGrader for this question.)
Exercise 4. Jordan form, decomposition
Find the Jordan-form of the following matrices.
A A
A A
(Write the Jordan form such that eigenvalues should be in ascending order of their absolute√

values. The absolute value of a complex number is defined as |a + bi| = a2 + b2)
Exercise 5. Function of matrices
Given

Find A10,A103, and eAt (when submitting to the Autograder, substitute t with 1).
Exercise 6. Diagonalization
Diagonlize the following matrix.
A
Such that A = MΛM−1
(Write the diagonal matrix Λ such that its eigenalues are in ascending order of their absolute values.)