Eigenvalue, Eigenvector Analysis of a Square Matrix A

Example 4.

`Example4.  Eigenvalue, eigenvector analysis of `*A*`:`

A = matrix([[-2, 2, 3], [-2, 3, 2], [-4, 2, 5]]), `  `*(A-I*lambda) = matrix([[-2-lambda, 2, 3], [-2, 3-lambda, 2], [-4, 2, 5-lambda]])

det(A-I*lambda)*` = `*(lambda-1)*(lambda-2)*(lambda-3) = 0

evecsAval1 = 1

evecsAmul1 = 1

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

lambda = 1

A-I*lambda = matrix([[-3, 2, 3], [-2, 2, 2], [-4, 2, 4]])

matrix([[-3, 2, 3], [-2, 2, 2], [-4, 2, 4]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

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

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

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

evecsAval2 = 2

evecsAmul2 = 1

evecsAvec2 = vector([1, 2, 0])

lambda = 2

A-I*lambda = matrix([[-4, 2, 3], [-2, 1, 2], [-4, 2, 3]])

matrix([[-4, 2, 3], [-2, 1, 2], [-4, 2, 3]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

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

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

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

evecsAval3 = 3

evecsAmul3 = 1

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

lambda = 3

A-I*lambda = matrix([[-5, 2, 3], [-2, 0, 2], [-4, 2, 2]])

matrix([[-5, 2, 3], [-2, 0, 2], [-4, 2, 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[3] = matrix([[1], [1], [1]])

`Use (column) eigenvectors of `*A*` to construct P:`

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

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

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

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