Finding the matrix determinant using Hessenberg decomposition

Here’s an example program in Java that uses Hessenberg decomposition to find the determinant of a matrix:

   import Jama.Matrix;

   public class MatrixDeterminant {

    public static void main(String[] args) {

        double[][] matrixData = {

            {1, 2, 3},

            {4, 5, 6},

            {7, 8, 9}

        };
        Matrix matrix = new Matrix(matrixData);

        double determinant = calculateDeterminant(matrix);

        System.out.println("Determinant: " + determinant);

    }

      public static double calculateDeterminant(Matrix matrix) {

        Matrix hessenberg = matrix.hessenberg();

        double determinant = 1.0;

       

        for (int i = 0; i < hessenberg.getRowDimension(); i++) {

            determinant *= hessenberg.get(i, i);

        }
        return determinant;

    }

}

This program uses the Jama library, which provides numerical linear algebra operations in Java. You can include the Jama library in your project by downloading the JAR file from the Jama project’s website and adding it to your project’s classpath.

In the example above, we create a 3×3 matrix with some sample data. We then calculate the determinant of the matrix by first obtaining its Hessenberg form using the `hessenberg()` method from the `Matrix` class. The Hessenberg form is an upper triangular matrix, and the determinant of an upper triangular matrix is simply the product of its diagonal elements. We iterate over the diagonal elements of the Hessenberg matrix and multiply them to calculate the determinant.

Note that you’ll need to download and include the Jama library in your project for the program to work correctly. You can find the Jama library and more information about it at https://math.nist.gov/javanumerics/jama/.