## 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.

**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] |
= | a_{11} |
a_{12} |
|||

a_{21} |
a_{22} |

and each eigenvector **v _{1}**,

**v**, takes the form

_{2}

(v) |
= | v_{1} |
||

v_{2} |

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).