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Engineering Assignment: Finding Eigenvalues and Eigenvectors for Given Matrices, Assignments of Electrical and Electronics Engineering

Rensselaer Polytechnic Institute (RPI)Electrical and Electronics Engineering

A university assignment from the department of electrical, computer, and systems engineering at rensselaer polytechnic institute, fall 2004. The assignment involves finding eigenvalues and eigenvectors for given matrices using various methods, including manual calculations and matlab. It also includes problems related to second-order systems and modal matrices.

Typology: Assignments

Pre 2010

Uploaded on 08/09/2009

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SCHOOLOFENGINEERING
DEPARTMENT OF ELECTRICAL,
COMPUTER,AND SYSTEMS ENGINEERING
S.A.T. Assignment #3 Fall 2004
Due: Thursday, September 23rd
1. Find the eigenvaluesand the eigenvectorsfor
A=[~ ~]. (lOpts)
2. Problem 2.8 a and b, page 64. (10pts)
(c) For the second-order system below,
i(t) =[~ ~]X(t) +[~]u(t)
find the response x(t) when x(0) =[-1 -1 fand u(t) = 1 for all tL 0 . You should be able to use
the results from Problem 2.8b. (10 pts)
3. For a certain A matrix,
14t +5t23t+2t2
2
eAt =II
. Find A. Hint: work with the derivative of eAt.
0 1+20t 16t (5 pts)
0-25t 1-20t
4. a.) Findthe eigenvalues and the eigenvectors for
2 -2 3
A=lll 1
1 3 -1
(15pts)
b.) Usingthe results frompart a.), find the modal matrix M. (5 pts)
c.) Verifythat M is the modal matrix by calculating M -1AM or by showing
AM =MA where A is a diagonal matrix with the eigenvalues of A on
the diagonal.
(5 pts)
Rensselaer Polytechnic Institute
110 8th Street I Troy, NY 12180-3590USA
Fax (518) 276-6261 or (518) 276-4403
- -- - --- -
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Download Engineering Assignment: Finding Eigenvalues and Eigenvectors for Given Matrices and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

SCHOOLOF ENGINEERING DEPARTMENT OF ELECTRICAL, COMPUTER,AND SYSTEMS ENGINEERING

S.A.T. Assignment #3 Fall 2004

Due: Thursday, September 23rd

  1. (^) Find the eigenvalues and the eigenvectors for

A=[~ ~].

(lOpts)

2. Problem 2.8 a and b, page 64. (10pts)

(c) For the second-order system below,

i(t) = [~ ~ ]X(t) + [~]u(t)

find the response x(t) when x (0) = [-1 -1 f and u(t) = 1 for all t L 0. You should be able to use the results from Problem 2.8b. (10 pts)

3. For a certain A matrix,

1 4t + 5 t 2 3 t + 2 t 2 2 eAt = I I

. Find A. Hint: work with the derivative of eAt. 0 1 +20t 16t (5 pts) 0 -25t 1 -20t

4. a.) Find the eigenvalues and the eigenvectors for

A=lll 1 1 3 -

(15 pts)

b.) Using the results from part a.), find the modal matrix M. (5 pts)

c.) Verify that M is the modal matrix by calculating M -1AM or by showing

AM = MA where A is a diagonal matrix with the eigenvalues of A on

the diagonal.

(5 pts)

Rensselaer Polytechnic Institute 110 8th Street I Troy,NY 12180-3590USA Fax (518) 276-6261 or (518) 276-

  • -- (^) - --- -
  1. (^) Repeat Problem 4 using MATLAB and the eig function. An example is shown below. (10 pts)

If A is real its eigenvalues can be found by using eig(A) which returns the eigenvalues in a column vector. The eigenvectors and the eigenvalues can be found with the double assignment statement, [x,d] = eig(A)

where the columns of x are the eigenvectors and the diagonal elements of d are the eigenvalues.

Example Consider the second-order linear system

[

XI(t)

]

= [

-4 1

][

XI(t)

xit) 0 -1 xit) ]

Use MATLAB to find the eigenvectors, the eigenvalues, and the modal matrix. This example was done by hand on page 93 of the text. The results can be verified using MATLAB.

»A=[-4 1;0 -1];

»[x1,d1]=eig(A) xl =

0

d1 =

»%Check M inverseAM. »L=inv(x1)* A*x L= -4. 0