Finding the bidiagonalization of a matrix.

Here’s an example of a Java program that finds the bidiagonalization of a given matrix:

   import java.util.Arrays;

   public class Bidiagonalization {

    public static void main(String[] args) {

        double[][] matrix = {

            {1, 2, 3},

            {4, 5, 6},

            {7, 8, 9}

        };

        int m = matrix.length;

        int n = matrix[0].length;

        double[][] bidiagonalMatrix = new double[m][n];

        // Perform bidiagonalization

        for (int i = 0; i < n; i++) {

            double[] column = getColumn(matrix, i);

            double norm = calculateNorm(column);

            if (norm != 0) {

                bidiagonalMatrix[0][i] = norm;

                double[] scaledColumn = scaleColumn(column, 1.0 / norm);

                bidiagonalMatrix[1][i] = dotProduct(scaledColumn, getColumn(matrix, i + 1));

                // Update the matrix by subtracting the projection

                matrix = subtractProjection(matrix, scaledColumn, i + 1);

            }

        }

        // Print the bidiagonal matrix

        for (double[] row : bidiagonalMatrix) {

            System.out.println(Arrays.toString(row));

        }

    }

    // Get a column from the matrix

    private static double[] getColumn(double[][] matrix, int colIndex) {

        int m = matrix.length;

        double[] column = new double[m];

        for (int i = 0; i < m; i++) {

            column[i] = matrix[i][colIndex];

        }

        return column;

    }

    // Calculate the Euclidean norm of a vector

    private static double calculateNorm(double[] vector) {

        double norm = 0;

        for (double value : vector) {

            norm += value * value;

        }

        return Math.sqrt(norm);

    }

    // Scale a column by a scalar value

    private static double[] scaleColumn(double[] column, double scalar) {

        int m = column.length;

        double[] scaledColumn = new double[m];

        for (int i = 0; i < m; i++) {

            scaledColumn[i] = column[i] * scalar;

        }

        return scaledColumn;

    }

    // Calculate the dot product of two vectors

    private static double dotProduct(double[] vector1, double[] vector2) {

        int m = vector1.length;

        double dotProduct = 0;

        for (int i = 0; i < m; i++) {

            dotProduct += vector1[i] * vector2[i];

        }

        return dotProduct;

    }

    // Subtract the projection of a vector onto another vector

    private static double[][] subtractProjection(double[][] matrix, double[] column, int colIndex) {

        int m = matrix.length;

        int n = matrix[0].length;

        double[][] updatedMatrix = new double[m][n];

        for (int i = 0; i < m; i++) {

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

                if (j == colIndex) {

                    updatedMatrix[i][j] = matrix[i][j];

                } else {

                    updatedMatrix[i][j] = matrix[i][j] - column[i] * matrix[i][colIndex];

                }

            }

        }

        return updatedMatrix;

    }

}

In this program, the `main` method takes a sample matrix and performs the bidiagonalization algorithm. The resulting bidiagonal matrix is then printed. The algorithm calculates the bidiagonalization by iterating

over the columns of the matrix, calculating the norm of each column, and finding the projection onto the next column. The matrix is then updated by subtracting the projection. Finally, the resulting bidiagonal matrix is printed to the console.