Eigenvalue, Eigenvector Analysis of a Square Matrix A

Example 3.

`Example3.  Eigenvalue, eigenvector analysis of `*A*`:`

A = matrix([[5, 6, 2], [0, -1, -8], [1, 0, -2]]), `  `*(A-I*lambda) = matrix([[5-lambda, 6, 2], [0, -1-lambda, -8], [1, 0, -2-lambda]])

det(A-I*lambda)*` = `*(lambda+4)*(lambda-3)^2 = 0

evecsAval1 = -4

evecsAmul1 = 1

evecsAvec1 = vector([-2, 8/3, 1])

lambda = -4

A-I*lambda = matrix([[9, 6, 2], [0, 3, -8], [1, 0, 2]])

matrix([[9, 6, 2], [0, 3, -8], [1, 0, 2]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, 2], [0, 1, -8/3], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a+2*c], [b-8/3*c], [0]]) = matrix([[0], [0], [0]])

v[1] = matrix([[-2], [8/3], [1]])

evecsAval2 = 3

evecsAmul2 = 2

evecsAvec2 = vector([5, -2, 1])

lambda = 3

A-I*lambda = matrix([[2, 6, 2], [0, -4, -8], [1, 0, -5]])

matrix([[2, 6, 2], [0, -4, -8], [1, 0, -5]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[1, 0, -5], [0, 1, 2], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])

matrix([[a-5*c], [b+2*c], [0]]) = matrix([[0], [0], [0]])

v[2] = matrix([[5], [-2], [1]])