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"
          Springer-Verlag, Berlin.

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

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

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


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