Finding the matrix rank using Hessenberg decomposition

Here’s a Java program that finds the rank of a matrix using the Hessenberg decomposition:

import java.util.Arrays;

public class MatrixRankHessenberg {   

    // Helper function to perform Hessenberg decomposition

    private static void hessenbergDecomposition(double[][] matrix, double[][] hessenberg) {

        int n = matrix.length;

        double[] ort = new double[n];       

        for (int k = 1; k < n - 1; k++) {

            double h = 0.0;         

            for (int i = k + 1; i < n; i++) {

                h += matrix[i][k - 1] * matrix[i][k - 1];

            }          

            if (h != 0.0) {

                double f = matrix[k + 1][k - 1];

                double g = Math.sqrt(h);

                if (f >= 0) {

                    g = -g;

                }

                h -= f * g;

                matrix[k + 1][k - 1] = f - g;              

                for (int i = k + 1; i < n; i++) {

                    ort[i] = matrix[i][k - 1] / h;

                }

               
                for (int j = k + 1; j < n; j++) {

                    double sum = 0.0;
                  
                    for (int i = k + 1; i < n; i++) {

                        sum += matrix[i][j] * ort[i];

                    }

                   
                    for (int i = k + 1; i < n; i++) {

                        matrix[i][j] -= sum * ort[i];

                    }

                }

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

                    double sum = 0.0;

                    for (int j = k + 1; j < n; j++) {

                        sum += matrix[i][j] * ort[j];

                    }

                    for (int j = k + 1; j < n; j++) {

                        matrix[i][j] -= sum * ort[j];

                    }

                }

            }

        }

        // Copy the upper triangular part of the matrix to the hessenberg matrix

        for (int i = 0; i < n; i++) {

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

                hessenberg[i][j] = (i <= j) ? matrix[i][j] : 0.0;

            }

        }

    } 

    // Helper function to count the number of non-zero rows

    private static int countNonZeroRows(double[][] matrix) {

        int count = 0;

       

        for (double[] row : matrix) {

            boolean isZeroRow = Arrays.stream(row).allMatch(value -> value == 0.0);

         
            if (!isZeroRow) {

                count++;

            }

        }

        return count;

    }

    // Function to find the rank of a matrix using Hessenberg decomposition

    public static int findMatrixRank(double[][] matrix) {

        int n = matrix.length;

        double[][] hessenberg = new double[n][n];

        // Perform Hessenberg decomposition

        hessenbergDecomposition(matrix, hessenberg);

        // Count the number of non-zero rows in the Hessenberg matrix

        int rank = countNonZeroRows(hessenberg);

        return rank;

    }

    // Test the program

    public static void main(String[] args) {

        // Example matrix

        double

[][] matrix = {

            {1, 2, 3},

            {4, 5, 6},

            {7, 8, 9}

        };

        int rank = findMatrixRank(matrix);

        System.out.println("Matrix rank: " + rank);

    }

}

In this program, the `hessenbergDecomposition` function performs the Hessenberg decomposition of the input matrix, and the `countNonZeroRows` function counts the number of non-zero rows in a matrix. The `findMatrixRank` function combines these two functions to find the rank of a matrix. In the `main` function, we provide an example matrix and print its rank. You can modify the `matrix` variable to test the program with different matrices.