#!/usr/bin/env python
# coding: utf-8

# # Orthogonal Iteration

# In[46]:


import numpy as np
import numpy.linalg as la

# Let's make a matrix with given eigenvalues:

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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.

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Q, R = la.qr(X)
X = A @ Q
print(Q)

# Now check that the (hopefully) converged $Q$ actually led to Schur form:

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la.norm(
    Q @ R @ Q.T
    - A)

# Do the eigenvalues match?

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R

# What are possible flaws in this plan?
# 
# * Will this always converge?
# * What about complex eigenvalues?