Finding the matrix determinant using Cholesky decomposition
To find the matrix determinant using Cholesky decomposition, we first need to check if the matrix is square and positive definite. If it satisfies these conditions, we can proceed with the Cholesky decomposition.
Here’s a Java program that implements this approach:
import java.util.Scanner;
public class CholeskyDeterminant {
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
System.out.print("Enter the size of the matrix: ");
int n = scanner.nextInt();
double[][] matrix = new double[n][n];
System.out.println("Enter the matrix elements row by row:");
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
matrix[i][j] = scanner.nextDouble();
}
}
double determinant = calculateDeterminant(matrix);
System.out.println("Determinant: " + determinant);
scanner.close();
}
public static double calculateDeterminant(double[][] matrix) {
int n = matrix.length;
// Check if the matrix is square
if (n != matrix[0].length) {
throw new IllegalArgumentException("Matrix is not square.");
}
// Check if the matrix is positive definite
for (int i = 0; i < n; i++) {
for (int j = 0; j < i; j++) {
if (matrix[i][j] != matrix[j][i]) {
throw new IllegalArgumentException("Matrix is not positive definite.");
}
}
if (matrix[i][i] <= 0) {
throw new IllegalArgumentException("Matrix is not positive definite.");
}
}
// Perform Cholesky decomposition
double[][] lower = new double[n][n];
for (int i = 0; i < n; i++) {
for (int j = 0; j <= i; j++) {
double sum = 0.0;
if (j == i) {
for (int k = 0; k < j; k++) {
sum += Math.pow(lower[j][k], 2);
}
lower[j][j] = Math.sqrt(matrix[j][j] - sum);
} else {
for (int k = 0; k < j; k++) {
sum += (lower[i][k] * lower[j][k]);
}
lower[i][j] = (matrix[i][j] - sum) / lower[j][j];
}
}
}
// Calculate determinant using the diagonal elements of the lower triangular matrix
double determinant = 1.0;
for (int i = 0; i < n; i++) {
determinant *= Math.pow(lower[i][i], 2);
}
return determinant;
}
}
In this program, we first prompt the user to enter the size of the matrix and then input the matrix elements row by row. The `calculateDeterminant` method checks if the matrix is square and positive definite. It performs the Cholesky decomposition to obtain the lower triangular matrix and calculates the determinant using the diagonal elements. Finally, the program outputs the determinant.
Note: This implementation assumes that the input matrix is a real symmetric positive definite matrix. It may not work correctly for matrices that don’t satisfy these conditions.