Finding the matrix rank using Bidiagonalization
Here’s an example of a Java program that calculates the rank of a matrix using the Bidiagonalization method:
public class MatrixRankCalculator {
public static void main(String[] args) {
double[][] matrix = {
{1, 2, 3},
{4, 5, 6},
{7, 8, 9}
};
int rank = calculateRank(matrix);
System.out.println("Matrix rank: " + rank);
}
public static int calculateRank(double[][] matrix) {
int m = matrix.length;
int n = matrix[0].length;
int rank = 0;
for (int i = 0; i < Math.min(m, n); i++) {
// Zero out elements below the diagonal
for (int j = i + 1; j < m; j++) {
double factor = matrix[j][i] / matrix[i][i];
for (int k = i; k < n; k++) {
matrix[j][k] -= factor * matrix[i][k];
}
}
// Zero out elements above the superdiagonal
if (i < n - 1) {
for (int j = i + 1; j < n; j++) {
double factor = matrix[i][j] / matrix[i][i + 1];
for (int k = i + 1; k < m; k++) {
matrix[k][j] -= factor * matrix[k][i + 1];
}
}
}
}
// Count the number of non-zero diagonal and superdiagonal elements
for (int i = 0; i < Math.min(m, n); i++) {
if (Math.abs(matrix[i][i]) > 1e-10) {
rank++;
}
if (i < n - 1 && Math.abs(matrix[i][i + 1]) > 1e-10) {
rank++;
}
}
return rank;
}
}
In this program, we define a `calculateRank` method that takes a 2D array representing a matrix and returns its rank. The Bidiagonalization algorithm is implemented within this method. The main logic involves zeroing out elements below the diagonal and above the superdiagonal using appropriate row operations. Then, the rank is determined by counting the number of non-zero diagonal and superdiagonal elements.
In the `main` method, we provide an example matrix and call the `calculateRank` method to compute and print the rank.
Note: The above code assumes that the matrix is a square matrix. If you want to handle rectangular matrices as well, you need to modify the code accordingly.