Finding the Schur decomposition of a matrix.

Here’s a Java program that finds the Schur decomposition of a matrix:

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

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

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

   public class SchurDecompositionExample {

    public static void main(String[] args) {

        // Define the matrix

        double[][] matrixData = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

        RealMatrix matrix = MatrixUtils.createRealMatrix(matrixData);

        // Compute the Schur decomposition

        EigenDecomposition decomposition = new EigenDecomposition(matrix);

        RealMatrix schurMatrix = decomposition.getT();

        RealMatrix unitaryMatrix = decomposition.getV();

        // Print the Schur matrix

        System.out.println("Schur Matrix:");

        printMatrix(schurMatrix);

        // Print the unitary matrix

        System.out.println("Unitary Matrix:");

        printMatrix(unitaryMatrix);

    }

    private static void printMatrix(RealMatrix matrix) {

        int rows = matrix.getRowDimension();

        int cols = matrix.getColumnDimension();

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

            for (int j = 0; j < cols; j++) {

                System.out.print(matrix.getEntry(i, j) + " ");

            }

            System.out.println();

        }

        System.out.println();

    }

}

In this example, we use the Apache Commons Math library to perform the Schur decomposition. The `EigenDecomposition` class provides the necessary methods to compute the Schur decomposition of a matrix. The program defines a sample matrix, computes its Schur decomposition, and then prints the Schur matrix and the unitary matrix.

Make sure you have the Apache Commons Math library added to your project dependencies in order to compile and run this program successfully.