Eigenvalues and Eigenvectors Calculator for a 2 X 2 Real Matrix
This page contains a routine that numerically finds the eigenvalues and eigenvectors of a 2 X 2 Real Matrix. The algorithm is from the EISPACK collection of subroutines.
Smith, B.T.; J.M. Boyle; J.J. Dongarra; B.S. Garbow; Y. Ikebe; V.C. Klema; and C.B. Moler.
"Matrix Eigensystem Routines--(EISPACK) Guide"
Garbow, B.S.; J.M. Boyle; J.J. Dongarra; and C.B. Moler.
"Matrix Eigensystem Routines--(EISPACK) Guide Extension"
λ is an eigenvalue (a scalar) of the Matrix [A] if there is a non-zero vector (v) such that the following relationship is satisfied:
[A](v) = λ (v)
Every vector (v) satisfying this equation is called an eigenvector of [A] belonging to the eigenvalue λ.
In the present case, we are dealing with a 2 X 2 Matrix:
and each eigenvector v1, v2, takes the form
To use this utility, you should have the a values ready to enter. If you have all the data ready, simply enter it, click the Solve button, and it will calculate the eigenvalues of [A] and the associated eigenvectors. Note that the a values are assumed to be real; however, the solutions may be complex. In other words, this utility calculates solutions that may have imaginary components (indicated by the "i"); however, it assumes the inputs are all real (it does not accept complex inputs).