I solved most of this assignment using Python, but not sure how to input (or even understand) the remaining information. Aren't you supposed to find the eigenvector given some eigenvalue, thus it appears in reverse in the following:
Let the symmetric 3 × 3 matrix A be A = ((1 2 1), (2 5 0), (1 0 5))
If A has the following three eigenvectors, then find three corresponding eigenvalues:
v1 = ((-5), (2), (1)); v2 = ((0), (-1), (2)); v3 = ((1), (2), (1))
Given a matrix arr and a vector vec, if vec is eigenvector of arr, then:
np.dot(arr, vec) == lambda_ * vec
therefore, going through all the values of np.dot(arr, vec) / vec (eventually ignoring the null vector as potential eigenvector, and suppressing errors for potential division by zero), would reveal the eigenvalue:
import numpy as np
def find_eigenvalue(arr, vec):
result = None
if not np.all(np.isclose(vec, 0.0)):
with np.errstate(divide='ignore', invalid='ignore'):
for x, y in zip(np.dot(arr, vec), vec):
if np.isclose(y, 0.0) and np.isclose(x, 0.0):
continue
if result is None:
result = x / y
elif not np.isclose(result, x / y):
result = None
break
return result
which works as expected:
print(find_eigenvalue(arr, np.array([0, 0, 0])))
# None
print(find_eigenvalue(arr, np.array([1, 0, 0])))
# None
print(find_eigenvalue(arr, np.array([0, 1, 0])))
# None
print(find_eigenvalue(arr, np.array([0, 0, 1])))
# None
print(find_eigenvalue(arr, np.array([0, -1, 2])))
# 5
print(find_eigenvalue(arr, np.array([-5, 2, 1])))
# -0.0
print(find_eigenvalue(arr, np.array([1, 2, 1])))
# 6.0
Finally, note that np.linalg.svd() will also compute all these:
u, s, vh = np.linalg.svd(arr)
and s will contain the eigenvalues, and u and vh will contain the eigenvectors (more precisely the left-singular eigenvectors in u and the hermitian of the right-singular eigenvectors in vh).