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# In[46]:
import numpy as np
import numpy.linalg as la
# Let's make a matrix with given eigenvalues:
# In[47]:
n = 5
np.random.seed(70)
eigvecs = np.random.randn(n, n)
eigvals = np.sort(np.random.randn(n))
A = np.dot(la.solve(eigvecs, np.diag(eigvals)), eigvecs)
print(eigvals)
# Let's make an array of iteration vectors:
# In[48]:
X = np.random.randn(n, n)
# Next, implement orthogonal iteration:
#
# * Orthogonalize.
# * Apply A
# * Repeat
#
# Run this cell in-place (Ctrl-Enter) many times.
# In[90]:
Q, R = la.qr(X)
X = A @ Q
print(Q)
# Now check that the (hopefully) converged $Q$ actually led to Schur form:
# In[91]:
la.norm(
Q @ R @ Q.T
- A)
# Do the eigenvalues match?
# In[92]:
R
# What are possible flaws in this plan?
#
# * Will this always converge?
# * What about complex eigenvalues?