Finding the matrix eigenvectors using Householder transformation
Here’s an example Java code that implements the Householder transformation to find the eigenvectors of a matrix:
import java.util.Arrays;
public class HouseholderEigenvectors {
public static void main(String[] args) {
double[][] matrix = {
{2, 1},
{1, 3}
};
double[][] eigenvectors = findEigenvectors(matrix);
System.out.println("Eigenvectors:");
for (double[] vector : eigenvectors) {
System.out.println(Arrays.toString(vector));
}
}
public static double[][] findEigenvectors(double[][] matrix) {
int n = matrix.length;
double[][] eigenvectors = new double[n][n];
double[][] identity = createIdentityMatrix(n);
for (int i = 0; i < n; i++) {
double[] x = Arrays.copyOf(matrix[i], n);
for (int j = 0; j < n; j++) {
if (j == i) {
eigenvectors[j] = Arrays.copyOf(identity[j], n);
} else {
double[] v = Arrays.copyOf(matrix[j], n);
double alpha = -v[i] / x[i];
double[] w = multiplyScalar(x, alpha);
v = addArrays(v, w);
matrix[j] = Arrays.copyOf(v, n);
eigenvectors[j] = addArrays(eigenvectors[j], multiplyScalar(eigenvectors[i], alpha));
}
}
}
return eigenvectors;
}
public static double[][] createIdentityMatrix(int n) {
double[][] identity = new double[n][n];
for (int i = 0; i < n; i++) {
identity[i][i] = 1.0;
}
return identity;
}
public static double[] multiplyScalar(double[] array, double scalar) {
int n = array.length;
double[] result = new double[n];
for (int i = 0; i < n; i++) {
result[i] = array[i] * scalar;
}
return result;
}
public static double[] addArrays(double[] array1, double[] array2) {
int n = array1.length;
double[] result = new double[n];
for (int i = 0; i < n; i++) {
result[i] = array1[i] + array2[i];
}
return result;
}
}
In this code, the `findEigenvectors` method implements the Householder transformation algorithm to compute the eigenvectors of the given matrix. It iteratively performs the transformation on the matrix and accumulates the eigenvectors as it progresses. The resulting eigenvectors are returned as a 2D array.
The main method demonstrates the usage of the `findEigenvectors` method by finding the eigenvectors of a sample 2×2 matrix. The resulting eigenvectors are then printed to the console.
Please note that this implementation assumes that the matrix is square and that the Householder transformation can be applied to it. Additionally, this code is a simplified version for educational purposes and may not handle all edge cases or optimizations