Eigenvalue, Eigenvector Analysis of a Square Matrix A

Example 5.

`Example5.  Eigenvalue, eigenvector analysis of `*A*`:`

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

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

evecsAval1 = 2

evecsAmul1 = 2

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

evecsAvec12 = vector([-1, 0, 1])

lambda = 2

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

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

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

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

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

evecsAval2 = 1

evecsAmul2 = 1

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

lambda = 1

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

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

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

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

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

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

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

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

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

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