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

# # 3x3 Givens Rotation
# 
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# 
# The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
# 
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
# THE SOFTWARE.
# </details>

# In[4]:


import numpy as np


# In[5]:


a = np.random.randn(3)
a


# Let's zero out $a_2$:

# In[7]:


G = np.zeros((3, 3))

c = a[0]/np.sqrt(a[0]**2 + a[1]**2)
s = a[1]/np.sqrt(a[0]**2 + a[1]**2)

G[:2,:2] = np.array([
        [c, s],
        [-s, c]
        ])
G


# Anything wrong with $G$?

# <!--
# G zeroes out the last component.
# -->
# (Edit this cell for solution.)

# How would we fix that issue?

# In[11]:


G[2,2] = 1


# In[10]:


G @ a


# Observations?