## Eigenvalues and Eigenvectors Calculator for a 3 X 3 Real Matrix |

This page contains a routine that numerically finds the eigenvalues and eigenvectors of a 3 X 3 **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, since we are dealing with a 3 X 3 Matrix and a 3-entry column vector,

a_{11} |
a_{12} |
a_{13} |
||||||

[A] |
= | a_{21} |
a_{22} |
a_{23} |
||||

a_{31} |
a_{32} |
a_{33} |

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

**v**,

_{2}**v**, etc takes the form

_{3}

v_{1} |
||||

(v) |
= | v_{2} |
||

v_{3} |

**HOW TO USE THIS UTILITY**

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).

Do not enter commas, brackets, etc. Also note that numbers in scientific notation are NOT recognized.

**IMPORTANT!** How to use the output.

If the i-th eigenvalue is real, the i-th COLUMN of the eigenvector Matrix contains the corresponding eigenvector.

If the i-th eigenvalue is complex with positive imaginary part, COLUMNS i and (i + 1) contain the real and imaginary parts of the corresponding eigenvector. The conjugate of this vector is the eigenvector for the conjugate eigenvalue.

Note the Error Code. If it does not equal -1, some eigenvalues and all eigenvectors are meaningless.

Error Code = -1: Normal completion.

Error Code > 0: If more than 30 iterations are required to determine an eigenvalue, the subroutine terminates. The Error Code gives the index of the eigenvalue for which the failure occurred. Eigenvalues λ _{ ErCode + 1 },
λ _{ ErCode + 2 }, . . . λ _{N} should be correct, but no eigenvectors are computed.