For a square matrix A, the statement "A has an eigenvalue Îģ" is equivalent to the statement "(AâÎģI)v=0 has a non-zero solution v". Say your friend claims that the matrix A= âââ91â12101â2â18ââ â has eigenvalue Îģ=8 and wants our help finding the eigenvector. We proceed by attempting to solve the system of linear equations (Aâ8I)v=0.
That is, we try to solve the system represented by the augmented matrix âââ11â1221â2â10âŖâŖâŖâŖ000ââ â. This matrix can be row-reduced to âââ100230â2â21âŖâŖâŖâŖ000ââ â. Recalling our experience in Math1131, we can see this has a unique solution, namely v=