Eigenvalue, Eigenvector Analysis of a Square Matrix A

Example 6 .

`Example6.

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

det(A-I*lambda)*` = `*(lambda-8)*(lambda-2)^2 = 0

evecsAval1 = 2

evecsAmul1 = 2

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

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

lambda = 2

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

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

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

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

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

evecsAval2 = 8

evecsAmul2 = 1

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

lambda = 8

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

matrix([[-4, 2, 2], [2, -4, 2], [2, 2, -4]])*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

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

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

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

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