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

a_{21} a_{22}

and each eigenvector **v**_{1}, **v**_{2}, takes the form

**(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).

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