Finding the matrix eigenvectors using inverse iteration

Here’s an example Java program that uses inverse iteration to find the eigenvectors of a matrix:

import Jama.Matrix;

import Jama.EigenvalueDecomposition;

public class MatrixEigenvectors {

    public static void main(String[] args) {

        // Define the matrix

        double[][] matrixData = {

            {2.0, -1.0, 0.0},

            {-1.0, 2.0, -1.0},

            {0.0, -1.0, 2.0}

        };

        Matrix matrix = new Matrix(matrixData);

        // Compute eigenvalues and eigenvectors

        EigenvalueDecomposition eig = matrix.eig();

        Matrix eigenvalues = eig.getD();

        Matrix eigenvectors = eig.getV();

        // Find eigenvectors using inverse iteration

        for (int i = 0; i < eigenvalues.getRowDimension(); i++) {

            double eigenvalue = eigenvalues.get(i, i);

            Matrix A = matrix.minus(Matrix.identity(matrix.getRowDimension(), matrix.getColumnDimension()).times(eigenvalue));

            // Initial guess for eigenvector

            Matrix x = Matrix.random(matrix.getColumnDimension(), 1);

            // Perform inverse iteration

            double tolerance = 1e-6; // Convergence tolerance

            int maxIterations = 100; // Maximum number of iterations

            int iteration = 0;

            double error = Double.POSITIVE_INFINITY;

            while (error > tolerance && iteration < maxIterations) {

                Matrix y = A.solve(x);

                double norm = y.norm2();

                x = y.times(1.0 / norm);

                error = A.times(x).minus(x.times(eigenvalue)).norm2();

                iteration++;

            }

            System.out.println("Eigenvalue: " + eigenvalue);

            System.out.println("Eigenvector: ");

            x.print(8, 4); // Print eigenvector with 8 decimal places

            System.out.println();

        }

    }

}

This program uses the Jama library (https://math.nist.gov/javanumerics/jama/) for matrix operations and the EigenvalueDecomposition class to compute the eigenvalues and eigenvectors of the matrix. The inverse iteration method is then applied to find the eigenvectors iteratively. The program prints the eigenvalues and corresponding eigenvectors for each eigenvalue found.