ערכים עצמיים: הכיוונים שאינם מסתובבים

function eigenvalues2x2(A) {
  const trace = A[0][0] + A[1][1];
  const determinant = A[0][0]*A[1][1] - A[0][1]*A[1][0];
  const discriminant = trace**2 - 4*determinant;
  if (discriminant < 0) {
    const real = trace/2;
    const imaginary = Math.sqrt(-discriminant)/2;
    return [{ real, imaginary }, { real, imaginary: -imaginary }];
  }
  const root = Math.sqrt(discriminant);
  return [(trace-root)/2, (trace+root)/2];
}

function matVec(A, v) {
  return A.map(row => row[0]*v[0] + row[1]*v[1]);
}

function residual(A, value, vector) {
  const image = matVec(A, vector);
  return image.map((entry, index) => entry-value*vector[index]);
}

const A = [[3, 1], [3, 5]];
console.log("eigenvalues", eigenvalues2x2(A));
console.log("λ=2 residual", residual(A, 2, [1, -1]));
console.log("λ=6 residual", residual(A, 6, [1, 3]));

from math import sqrt

def eigenvalues_2x2(A):
    trace = A[0][0] + A[1][1]
    determinant = A[0][0]*A[1][1] - A[0][1]*A[1][0]
    discriminant = trace**2 - 4*determinant
    if discriminant < 0:
        real = trace/2
        imaginary = sqrt(-discriminant)/2
        return [(real, imaginary), (real, -imaginary)]
    root = sqrt(discriminant)
    return [(trace-root)/2, (trace+root)/2]

def mat_vec(A, v):
    return [sum(a*b for a, b in zip(row, v)) for row in A]

def residual(A, value, vector):
    image = mat_vec(A, vector)
    return [entry-value*v for entry, v in zip(image, vector)]

A = [[3, 1], [3, 5]]
print("eigenvalues", eigenvalues_2x2(A))
print("lambda=2 residual", residual(A, 2, [1, -1]))
print("lambda=6 residual", residual(A, 6, [1, 3]))