## Eigenvalues and Eigenvectors of a 2 X 2 Matrix, a Numerical Solution Utility

This page contains a routine that numerically finds the eigenvalues and eigenvectors of a 2 X 2 Real Matrix. It is written in JavaScript, so make sure that JavaScript is enabled in your browser. The algorithm is from the EISPACK collection of subroutines.

References:

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"
Springer-Verlag, Berlin.
1976

Garbow, B.S.; J.M. Boyle; J.J. Dongarra; and C.B. Moler.
"Matrix Eigensystem Routines--(EISPACK) Guide Extension"
Springer-Verlag, Berlin.
1977

The original sub-routines were written in FORTRAN and have been translated to Javascript here. Although all care has been taken to ensure that the sub-routines were translated accurately, some errors may have crept into the translation. These errors are mine; the original FORTRAN routines have been thoroughly tested and work properly. Please report any errors to the webmaster.

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,

[A]  =   a11   a12
a21   a22

and each eigenvector v1, v2, takes the form

(v)   =   v1
v2

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).

 a11 a12 a21 a22

 λ1 = + i
The associated eigenvector is:
 v1 = + i + i
 λ2 = + i
The associated eigenvector is:
 v2 = - i - i