Eigenvalue, Eigenvector Analysis of a Square Matrix A

Example 1.

`Example1.  Eigenvalue, eigenvector analysis of `*A*`:`

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

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

evecsAval1 = 1

evecsAmul1 = 1

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

lambda = 1

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

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

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

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

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

evecsAval2 = 2

evecsAmul2 = 1

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

lambda = 2

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

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

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

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

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

evecsAval3 = 3

evecsAmul3 = 1

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

lambda = 3

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

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

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

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

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