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445/545
TEST # 3 (Final Exam)
12/11/01
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TEST # 3 (Final Exam)
12/17/02 Page 1
Problem #1:
Let A be a symmetric 3x3 matrix.
(a)-[3 pts.]- Determine the distinct eigenvalues and their algebraic multiplicities of A .
(b)-[4 pts.]-Using the result of part (a) find 3 linearly independent eigenvectors of A .
(c)-[4 pts.]-Calculate P 1 AP where the columns of P are the eigenvectors found in part (b).
1
P APt
.
(d)-[4pts.]- Use the result of part (c) to calculate e
P 1 APt -1
P  e At .
(e)-[5 pts.]- Use the result of part (d) to calculate Pe
(f)-[5 pts.]- Use the result of part (e) to calculate the solution of the initial value problem
x  Ax, x(0)  x 0 . Your answer should be expressed as a linear combination of the columns of e At .
Problem #2:
Let A be a 3x3 matrix with real and complex eigenvalues.
(a)-[4 pts.]- Determine the distinct eigenvalues of the matrix A .
(b)-[5 pts.]-Using the result of part (a) find 3 linearly independent real valued eigenvectors of A .
(c)-[4 pts.]-Calculate P 1 AP where the columns of P are the eigenvectors found in part (b).
P 1 APt
.
(d)-[4pts.]- Use the result of part (c) to calculate e
1
P APt -1
P  e At .
(e)-[4 pts.]- Use the result of part (d) to calculate Pe
(f)-[4 pts.]- Use the result of part (e) to calculate the solution of the initial value problem
x  Ax, x(0)  x o  given .
Problem #3:
Let A be a 4x4 matrix with one real eigenvalue.
(a)-[3 pts.]- Determine the eigenvalue of the matrix A .
(b)-[8 pts.]-Using the result of part (a) find 4 linearly independent vectors consisting of eigenvectors
and generalized eigenvectors of A .
(c)-[14 pts.]-Calculate e At u 1 , e At u 2 , e At u 3 , e At u 4 , where u1 , u 2 , u 3 , u 4 are the 4 linearly independent vectors
found in part (b).
Problem #4:
Let A be a 3x3 matrix with real and complex eigenvalues.
(a)-[4 pts.]- Calculate the eigenvalues of the matrix. A .
(b)-[12 pts.]-Using the result of part (a) find 3 linearly independent real valued eigenvectors of A .
(c)-[9 pts.]-Use the results of parts (a) and (b) to find E s , E u , and E c , the stable, unstable and center
subspaces of the system x  Ax .
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Problem #5:
(a)-[25 pts.]- Solve the following initial value problem: x  Ax  y, x(0)  ( x10 , x20 )T .
 y 1 ( t )
where A is a constant 2x2 matrix and y  
 is a given vector valued function.
 y 2 ( t )
Carry out all calculations.
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TEST #3 (Final Exam)
TEST # 3 (Final Exam)
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Problem #6:
Given a 2-dimensional non-linear system x  f ( x , y ), y  g( x , y ) with an equilibrium point at (0, 0).
(a)-[6 pts.]-Find V ( x , y ) such that V ( 0 , 0 )  0 and x  V x ( x , y )  f ( x , y ), y  V y ( x , y )  g( x , y ) .
(b)-[8 pts.]-Demonstrate that V ( x , y ) is a Liapunov function for the given system at (0, 0) and use this
function to determine the stability of (0, 0).
(c)-[6 pts.]- Verify that function V ( x , y ) found in part (a) is a Hamiltonian for the system
x   g( x , y ), y  f (x, y) and use this function to classify the equilibrium point (0, 0).
Also use V ( x , y ) to describe the orbits of the Hamiltonian system near (0, 0).
(d)-[5 pts.]-What is the relationship between the orbits of the gradient system and the orbits of
the Hamiltonian system?
Problem #7:
Consider the 2-dimensional non-linear system x  f ( x , y ), y  g( x , y ) .
(a)-[15 pts.]-Find a Liapunov function for the given system at (0, 0).
Hint: Try V ( x , y )  ax 2  by 2 where a and b are constants.
(b)-[10 pts.]-Use the result of (b) to classify the equilibrium point at (0, 0). Is it stable or unstable?
Is it asymptotically stable or unstable?
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TEST # 3 (Final Exam)
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