Finding the singular value decomposition of a rectangular matrix

Here’s an example of a Java problem that demonstrates how to find the singular value decomposition (SVD) of a rectangular matrix using the Apache Commons Math library:

   import org.apache.commons.math3.linear.*;

   public class SVDExample {

    public static void main(String[] args) {

        // Create a sample rectangular matrix

        double[][] matrixData = {

                {1, 2, 3},

                {4, 5, 6}

        };

        // Create a RealMatrix object from the matrix data

        RealMatrix matrix = MatrixUtils.createRealMatrix(matrixData);

        // Perform Singular Value Decomposition

        SingularValueDecomposition svd = new SingularValueDecomposition(matrix);

        // Get the matrices from the SVD

        RealMatrix uMatrix = svd.getU();

        RealMatrix sMatrix = svd.getS();

        RealMatrix vMatrix = svd.getV();

        // Print the matrices

        System.out.println("U matrix:");

        System.out.println(uMatrix);

        System.out.println("S matrix:");

        System.out.println(sMatrix);

        System.out.println("V matrix:");

        System.out.println(vMatrix);

    }

}

In this example, we create a rectangular matrix with dimensions 2×3 and find its singular value decomposition using the `SingularValueDecomposition` class from the Apache Commons Math library. The resulting U, S, and V matrices are then printed.

Make sure to have the Apache Commons Math library added to your project’s dependencies in order to run this code successfully.