Eigenvalues ONLY Calculator for a 3 x 3 Real Symmetric Matrix |
This page contains a routine that numerically finds the eigenvalues ONLY of a 3 x 3 Real, Symmetric 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.
As an example, in the case of a 3 x 3 Matrix and a 3-entry column vector,
| a11 | a12 | a13 | ||||||
| [A] | = | a21 | a22 | a23 | ||||
| a31 | a32 | a33 |
and each eigenvector v takes the form
| v1 | ||||
| (v) | = | v2 | ||
| v3 |
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]. Note that the a values are assumed to be real; this utility does not accept complex inputs.
Taking advantage of the fact that [A] is symmetric, the user only has to input a values for entries along the diagonal and below the diagonal.
Do not enter commas, brackets, etc. Also note that numbers in scientific notation are NOT recognized.
Because the matrix is symmetric, the eigenvalues are real (assuming they can be computed).
IMPORTANT!
Note the Error Code. If it does not equal 0, some eigenvalues may not have been computed.
Enter the [A] values:
If 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 1, λ 2, . . . λ ErCode - 1 should be correct.