Method of Finding Eigenvalues and Eigenvectors
Method of Finding Eigenvalues and Eigenvectors: Let A be an n × n matrix:
1. To find the eigen value of A solve the characteristics equations
This is a polynomial equation in λ of degree n. We only consider real roots of this equations, in the class.
2. Given an eigenvalue λi to find the eigenspace E(λi) corresponding to λi, we solve the linear system
As usual to solve this we reduce it to the row echelon form or Gauss-Jordan form. Since λi is an eigenvalue, at least one row of the echelon form will be zero.
1. Verify that λ1 = 5 is a eigenvalue of A and X1 = (1,2,-1)T is a corresponding eigenvector.
Solution: We need to check AX1 = 5X1, we have
So assertion is verified.
2. Verify that λ2 = - 3 is a eigenvalue of A and x2 = (-2,1,0)T is corresponding eigenvector.
Solution: We need to check AX2 = - 3X2, we have