Finding the matrix rank using LU decomposition.

Here’s a Java program that uses LU decomposition to find the rank of a matrix:

import java.util.Arrays;

public class MatrixRankLU {

    public static void main(String[] args) {

        double[][] matrix = {

                {1, 2, 3},

                {4, 5, 6},

                {7, 8, 9}

        };

        int rank = findRank(matrix);

        System.out.println("Rank of the matrix: " + rank);

    }

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

        int rowCount = matrix.length;

        int colCount = matrix[0].length;

        int rank = Math.min(rowCount, colCount);

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

            if (matrix[i][i] != 0) {

                // Perform LU decomposition

                for (int j = i + 1; j < rowCount; j++) {

                    double factor = matrix[j][i] / matrix[i][i];

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

                        matrix[j][k] -= factor * matrix[i][k];

                    }

                }

            } else {

                boolean reduced = false;

                // Swap rows to get a non-zero pivot

                for (int j = i + 1; j < rowCount; j++) {

                    if (matrix[j][i] != 0) {

                        double[] temp = matrix[i];

                        matrix[i] = matrix[j];

                        matrix[j] = temp;

                        reduced = true;

                        break;

                    }

                }

                if (!reduced) {

                    rank--;

                    for (int j = i; j < rowCount - 1; j++) {

                        matrix[j] = matrix[j + 1];

                    }

                }

                i--;

            }

        }

        return rank;

    }

}

This program demonstrates how to find the rank of a matrix using LU decomposition. The `findRank` method takes a 2D array representing the matrix as input and returns its rank as an integer.

In the `main` method, we create a sample matrix and call the `findRank` method to compute its rank. Finally, the program prints the rank to the console.

Note that this implementation assumes that the input matrix is a rectangular matrix with non-zero dimensions. If you plan to use it with irregular or zero-sized matrices, additional checks and error handling should be added.