Finding the matrix condition number using Schur decomposition.

Here’s an example program in Java that uses Schur decomposition to find the condition number of a matrix:

import org.apache.commons.math3.linear.Array2DRowRealMatrix;

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

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

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

public class MatrixConditionNumber {

    public static void main(String[] args) {

        // Define your matrix

        double[][] matrixData = {

                {1.0, 2.0, 3.0},

                {4.0, 5.0, 6.0},

                {7.0, 8.0, 9.0}

        };

        Array2DRowRealMatrix matrix = new Array2DRowRealMatrix(matrixData);

        // Compute Schur decomposition

        EigenDecomposition decomposition = new EigenDecomposition(matrix);

        Array2DRowRealMatrix t = new Array2DRowRealMatrix(decomposition.getT().getData());

        // Compute the inverse of T matrix

        Array2DRowRealMatrix inverseT = MatrixUtils.inverse(t);

        // Compute the condition number

        double conditionNumber = t.getNorm() * inverseT.getNorm();

        System.out.println("Condition Number: " + conditionNumber);

    }

}

Make sure you have the Apache Commons Math library (`org.apache.commons.math3`) added to your project dependencies. This library provides the necessary classes for matrix operations and decompositions.

In the code, you first define your matrix using a 2D array. Then, you compute the Schur decomposition of the matrix using the `EigenDecomposition` class from Apache Commons Math. Next, you extract the upper triangular matrix `T` from the decomposition and compute its inverse. Finally, you calculate the condition number as the product of the norms of `T` and its inverse.

Remember to adjust the matrix dimensions and data according to your specific requirements.