Name ____Answers_____________                  Linear Algebra, Quiz 3, Summer 2004  

 

Problem 1.  Do a complete eigenvalue, eigenvector analysis of the matrix A, including finding the matrix P that diagonalizes A and using P to diagonalize A.

`Problem 1.  Eigenvalue, eigenvector analysis of `*A*`:`

A = matrix([[3, 1, 0], [4, 0, 0], [1, -3, 3]]), `  `*(A-I*lambda) = matrix([[3-lambda, 1, 0], [4, -lambda, 0], [1, -3, 3-lambda]])

det(A-I*lambda)*` = `*(lambda-3)*(lambda-4)*(lambda+1) = 0

evecsAval1 = 4

evecsAmul1 = 1

evecsAvec1 = vector([1, 1, -2])

lambda = 4

A-I*lambda = matrix([[-1, 1, 0], [4, -4, 0], [1, -3, -1]])

matrix([[-1, 1, 0], [4, -4, 0], [1, -3, -1]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, 1/2], [0, 1, 1/2], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a+1/2*c], [b+1/2*c], [0]]) = matrix([[0], [0], [0]])

v[1] = matrix([[1], [1], [-2]])

evecsAval2 = 3

evecsAmul2 = 1

evecsAvec2 = vector([0, 0, 1])

lambda = 3

A-I*lambda = matrix([[0, 1, 0], [4, -3, 0], [1, -3, 0]])

matrix([[0, 1, 0], [4, -3, 0], [1, -3, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, 0], [0, 1, 0], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a], [b], [0]]) = matrix([[0], [0], [0]])

v[2] = matrix([[0], [0], [1]])

evecsAval3 = -1

evecsAmul3 = 1

evecsAvec3 = vector([1, -4, -13/4])

lambda = -1

A-I*lambda = matrix([[4, 1, 0], [4, 1, 0], [1, -3, 4]])

matrix([[4, 1, 0], [4, 1, 0], [1, -3, 4]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, 4/13], [0, 1, -16/13], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a+4/13*c], [b-16/13*c], [0]]) = matrix([[0], [0], [0]])

v[3] = matrix([[1], [-4], [-13/4]])

`A is diagonalizable using the eigenvectors of `*A*` to construct P:`

v[1] = matrix([[1], [1], [-2]]), v[2] = matrix([[0], [0], [1]]), v[3] = matrix([[1], [-4], [-13/4]])

P = matrix([[1, 0, 1], [1, 0, -4], [-2, 1, -13/4]]), P^`-1` = matrix([[4/5, 1/5, 0], [9/4, -1/4, 1], [1/5, -1/5, 0]])

P^`-1`*P*` = `*matrix([[4/5, 1/5, 0], [9/4, -1/4, 1], [1/5, -1/5, 0]])*matrix([[1, 0, 1], [1, 0, -4], [-2, 1, -13/4]]) = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

P^`-1`*A*P*` = `*matrix([[4/5, 1/5, 0], [9/4, -1/4, 1], [1/5, -1/5, 0]])*matrix([[3, 1, 0], [4, 0, 0], [1, -3, 3]])*matrix([[1, 0, 1], [1, 0, -4], [-2, 1, -13/4]]) = matrix([[4, 0, 0], [0, 3, 0], [0, 0...

Problem 2.  Find a matrix A that has the following eigenvalues and corresponding eigenvectors. Make sure to show all your work and explain your reasoning.

lambda[1] = 0, lambda[2] = 1, lambda[3] = -1, v[1] = matrix([[0], [1], [-1]]), v[2] = matrix([[1], [-1], [1]]), v[3] = matrix([[0], [1], [1]])

v[1] = matrix([[0], [1], [-1]]), v[2] = matrix([[1], [-1], [1]]), v[3] = matrix([[0], [1], [1]])

P = matrix([[0, 1, 0], [1, -1, 1], [-1, 1, 1]]), P^`-1` = matrix([[1, 1/2, -1/2], [1, 0, 0], [0, 1/2, 1/2]]), D = matrix([[0, 0, 0], [0, 1, 0], [0, 0, -1]])

P^`-1`*A*P = D

A = P*D*P^`-1`

A*` = `*matrix([[0, 1, 0], [1, -1, 1], [-1, 1, 1]])*matrix([[0, 0, 0], [0, 1, 0], [0, 0, -1]])*matrix([[1, 1/2, -1/2], [1, 0, 0], [0, 1/2, 1/2]]) = matrix([[1, 0, 0], [-1, -1/2, -1/2], [1, -1/2, -1/2]]...

Problem 2 Check:  

`Problem 2.  Eigenvalue, eigenvector analysis of `*A*`:`

A = matrix([[1, 0, 0], [-1, -1/2, -1/2], [1, -1/2, -1/2]]), `  `*(A-I*lambda) = matrix([[1-lambda, 0, 0], [-1, -1/2-lambda, -1/2], [1, -1/2, -1/2-lambda]])

det(A-I*lambda)*` = `*(lambda-1)*lambda*(lambda+1) = 0

evecsAval1 = 0

evecsAmul1 = 1

evecsAvec1 = vector([0, -1, 1])

lambda = 0

A-I*lambda = matrix([[1, 0, 0], [-1, -1/2, -1/2], [1, -1/2, -1/2]])

matrix([[1, 0, 0], [-1, -1/2, -1/2], [1, -1/2, -1/2]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, 0], [0, 1, 1], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a], [b+c], [0]]) = matrix([[0], [0], [0]])

v[1] = matrix([[0], [-1], [1]])

evecsAval2 = 1

evecsAmul2 = 1

evecsAvec2 = vector([1, -1, 1])

lambda = 1

A-I*lambda = matrix([[0, 0, 0], [-1, -3/2, -1/2], [1, -1/2, -3/2]])

matrix([[0, 0, 0], [-1, -3/2, -1/2], [1, -1/2, -3/2]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, -1], [0, 1, 1], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a-c], [b+c], [0]]) = matrix([[0], [0], [0]])

v[2] = matrix([[1], [-1], [1]])

evecsAval3 = -1

evecsAmul3 = 1

evecsAvec3 = vector([0, 1, 1])

lambda = -1

A-I*lambda = matrix([[2, 0, 0], [-1, 1/2, -1/2], [1, -1/2, 1/2]])

matrix([[2, 0, 0], [-1, 1/2, -1/2], [1, -1/2, 1/2]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, 0], [0, 1, -1], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a], [b-c], [0]]) = matrix([[0], [0], [0]])

v[3] = matrix([[0], [1], [1]])

`A is diagonalizable using the eigenvectors of `*A*` to construct P:`

v[1] = matrix([[0], [-1], [1]]), v[2] = matrix([[1], [-1], [1]]), v[3] = matrix([[0], [1], [1]])

P = matrix([[0, 1, 0], [-1, -1, 1], [1, 1, 1]]), P^`-1` = matrix([[-1, -1/2, 1/2], [1, 0, 0], [0, 1/2, 1/2]])

P^`-1`*P*` = `*matrix([[-1, -1/2, 1/2], [1, 0, 0], [0, 1/2, 1/2]])*matrix([[0, 1, 0], [-1, -1, 1], [1, 1, 1]]) = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

P^`-1`*A*P*` = `*matrix([[-1, -1/2, 1/2], [1, 0, 0], [0, 1/2, 1/2]])*matrix([[1, 0, 0], [-1, -1/2, -1/2], [1, -1/2, -1/2]])*matrix([[0, 1, 0], [-1, -1, 1], [1, 1, 1]]) = matrix([[0, 0, 0], [0, 1, 0], [...