Eigenvalues and Eigenvectors

How to find Eigenvalues. Matrix = A. Arbitrary variable = k. Determinant = Det. Identity matrix = I.

Det(A - kI)

Get Det and solve for k when it is equal to zero. The answer is the eigenvalue. The answer to this are the only eigenvalues of matrix A.

When there are multiple eigenvalues of the same value, it is only counted as 1. eg: if there are eigenvalues 1, 1 and 2, there are only two eigenvalues of 1 and 2. We say eigenvalue 1 has a multiplicity of 2.

How to find Eigenvectors.

Solve for (A- kI)x = 0 for solutions other than the zero vector. The value k is the eigenvectors that are calculated previously(above).

You need eigenvalues to calculate the eigenvectors. Any scalar multiple of the eigenvector is a valid eigenvector of eigenvalue k.

http://math.stackexchange.com/questions/243533/how-to-intuitively-understand-eigenvalue-and-eigenvector

https://en.wikipedia.org/wiki/Invariant_(mathematics)

https://en.wikipedia.org/wiki/Linear_map