EigenOps
org.ejml.ops

## Class EigenOps

• ```public class EigenOps
extends Object```
Additional functions related to eigenvalues and eigenvectors of a matrix.
• ### Constructor Summary

Constructors
Constructor and Description
`EigenOps()`
• ### Method Summary

Methods
Modifier and Type Method and Description
`static double[]` ```boundLargestEigenValue(DenseMatrix64F A, double[] bound)```
Generates a bound for the largest eigen value of the provided matrix using Perron-Frobenius theorem.
`static double` ```computeEigenValue(DenseMatrix64F A, DenseMatrix64F eigenVector)```
Given matrix A and an eigen vector of A, compute the corresponding eigen value.
`static Eigenpair` ```computeEigenVector(DenseMatrix64F A, double eigenvalue)```
Given an eigenvalue it computes an eigenvector using inverse iteration:
for i=1:MAX {
(A - μI)z(i) = q(i-1)
q(i) = z(i) / ||z(i)||
λ(i) = q(i)T A q(i)
}
`static DenseMatrix64F` `createMatrixD(EigenDecomposition eig)`
A diagonal matrix where real diagonal element contains a real eigenvalue.
`static DenseMatrix64F` `createMatrixV(EigenDecomposition<DenseMatrix64F> eig)`
Puts all the real eigenvectors into the columns of a matrix.
`static Eigenpair` `dominantEigenpair(DenseMatrix64F A)`
Computes the dominant eigen vector for a matrix.
• ### Methods inherited from class java.lang.Object

`equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ### Constructor Detail

• #### EigenOps

`public EigenOps()`
• ### Method Detail

• #### computeEigenValue

```public static double computeEigenValue(DenseMatrix64F A,
DenseMatrix64F eigenVector)```

Given matrix A and an eigen vector of A, compute the corresponding eigen value. This is the Rayleigh quotient.

xTAx / xTx

Parameters:
`A` - Matrix. Not modified.
`eigenVector` - An eigen vector of A. Not modified.
Returns:
The corresponding eigen value.
• #### computeEigenVector

```public static Eigenpair computeEigenVector(DenseMatrix64F A,
double eigenvalue)```

Given an eigenvalue it computes an eigenvector using inverse iteration:
for i=1:MAX {
(A - μI)z(i) = q(i-1)
q(i) = z(i) / ||z(i)||
λ(i) = q(i)T A q(i)
}

NOTE: If there is another eigenvalue that is very similar to the provided one then there is a chance of it converging towards that one instead. The larger a matrix is the more likely this is to happen.

Parameters:
`A` - Matrix whose eigenvector is being computed. Not modified.
`eigenvalue` - The eigenvalue in the eigen pair.
Returns:
The eigenvector or null if none could be found.
• #### dominantEigenpair

`public static Eigenpair dominantEigenpair(DenseMatrix64F A)`

Computes the dominant eigen vector for a matrix. The dominant eigen vector is an eigen vector associated with the largest eigen value.

WARNING: This function uses the power method. There are known cases where it will not converge. It also seems to converge to non-dominant eigen vectors some times. Use at your own risk.

Parameters:
`A` - A matrix. Not modified.
• #### boundLargestEigenValue

```public static double[] boundLargestEigenValue(DenseMatrix64F A,
double[] bound)```

Generates a bound for the largest eigen value of the provided matrix using Perron-Frobenius theorem. This function only applies to non-negative real matrices.

For "stochastic" matrices (Markov process) this should return one for the upper and lower bound.

Parameters:
`A` - Square matrix with positive elements. Not modified.
`bound` - Where the results are stored. If null then a matrix will be declared. Modified.
Returns:
Lower and upper bound in the first and second elements respectively.
• #### createMatrixD

`public static DenseMatrix64F createMatrixD(EigenDecomposition eig)`

A diagonal matrix where real diagonal element contains a real eigenvalue. If an eigenvalue is imaginary then zero is stored in its place.

Parameters:
`eig` - An eigenvalue decomposition which has already decomposed a matrix.
Returns:
A diagonal matrix containing the eigenvalues.
• #### createMatrixV

`public static DenseMatrix64F createMatrixV(EigenDecomposition<DenseMatrix64F> eig)`

Puts all the real eigenvectors into the columns of a matrix. If an eigenvalue is imaginary then the corresponding eigenvector will have zeros in its column.

Parameters:
`eig` - An eigenvalue decomposition which has already decomposed a matrix.
Returns:
An m by m matrix containing eigenvectors in its columns.