Finding the matrix eigenvalues using Householder transformation

Here’s a Java code snippet that demonstrates how to find the eigenvalues of a matrix using Householder transformation:

import java.util.Arrays;

public class HouseholderEigenvalues {

    // Helper function to print a matrix

    private static void printMatrix(double[][] matrix) {

        for (double[] row : matrix) {

            System.out.println(Arrays.toString(row));

        }

        System.out.println();

    }
    // Helper function to compute the dot product of two vectors

    private static double dotProduct(double[] vector1, double[] vector2) {

        double result = 0.0;

        for (int i = 0; i < vector1.length; i++) {

            result += vector1[i] * vector2[i];

        }

        return result;

    }
    // Helper function to apply Householder transformation to a matrix

    private static void applyHouseholderTransformation(double[][] matrix, double[] v) {

        int n = matrix.length;

        double[] w = new double[n];

        for (int j = 0; j < n; j++) {

            double dot = 0.0;

            for (int i = 0; i < n; i++) {

                dot += matrix[i][j] * v[i];

            }

            for (int i = 0; i < n; i++) {

                w[i] += dot * v[i];

            }

        }

        for (int j = 0; j < n; j++) {

            for (int i = 0; i < n; i++) {

                matrix[i][j] -= 2.0 * v[i] * w[j];

            }

        }

    }

    // Function to compute the eigenvalues of a matrix using Householder transformation

    public static double[] findEigenvalues(double[][] matrix) {

        int n = matrix.length;

        double[] eigenvalues = new double[n];

        for (int k = n - 1; k >= 1; k--) {

            double[] v = new double[n];

            for (int i = 0; i < k; i++) {

                v[i] = matrix[i][k];

            }

            v[k] = 1.0;          

            applyHouseholderTransformation(matrix, v);

            double sum = 0.0;

            for (int i = 0; i < k; i++) {

                sum += matrix[i][k] * matrix[i][k];

            }         

            double norm = Math.sqrt(sum);

            double alpha = -Math.signum(matrix[k][k]) * norm;

            for (int i = 0; i < k; i++) {

                matrix[i][k] /= alpha;

            }

            matrix[k][k] -= 1.0;

            for (int j = 0; j < k; j++) {

                matrix[k][j] = 0.0;

            }

        }

        for (int i = 0; i < n; i++) {

            eigenvalues[i] = matrix[i][i];

        }

        return eigenvalues;

    }

    public static void main(String[] args) {

        double[][] matrix = {

            {1.0, 2.0, 3.0},

            {4.0, 5.0, 6.0},

            {7.0, 8.0, 9.0}

        };

        System.out.println("Original Matrix:");

        printMatrix(matrix);

        double[] eigenvalues = findEigenvalues(matrix);      

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

        System.out.println(Arrays.toString(eigen

values));

    }

}

In this code, the `findEigenvalues` method implements the Householder transformation algorithm to compute the eigenvalues of a given matrix. The `applyHouseholderTransformation` method applies the Householder transformation to the matrix. The `dotProduct` method computes the dot product of two vectors. The `printMatrix` method is a helper function to print the matrix.

In the `main` method, a sample matrix is defined and its eigenvalues are computed using the `findEigenvalues` method. The resulting eigenvalues are then printed to the console.

Note: This code assumes that the input matrix is square.