# 15 Pre-Class Assignment: Diagonalization and Powers¶

## 1. Eigenvalues and eigenvectors review¶

Definition: A non-zero vector $$x$$ in $$R^n$$ is called an eigenvector of a $$n\times n$$ matrix $$A$$ if $$Ax$$ is a scalar multiple of $$x$$. If $$Ax = \lambda x$$, then $$\lambda$$ is called the eigenvalue of $$A$$ corresponding to $$x$$.

### Steps for finding the eigenvalues and eigenvectors¶

We want to find $$\lambda$$ and non-zero vector $$x$$ such that $$Ax=\lambda x$$ for a $$n\times n$$ matrix.

1. We introduce an identity matrix $$I$$ of $$n\times n$$. Then the equation becomes $$$Ax = \lambda I x$$$$Ax-\lambda I x = 0$$$$(A-\lambda I)x = 0$$$

2. This suggests that we want to find $$\lambda$$ such that $$(A-\lambda I)x=0$$ has a non-trivial solution. It is equivalent to that the matrix $$A-\lambda I$$ is singular, i.e., has a determinant of $$0$$. $$$|A-\lambda I|=0$$$

3. The determinant is polynomial in $$\lambda$$ (called the characteristic polynomial of $$A$$) with degree $$n$$. We solve this equation (called the characteristic equation) for all possible $$\lambda$$ (eigenvalues).

4. After finding the eigenvalues, we substitute them back into $$$(A-\lambda I)x=0$$$$and find the eigenvectors$$x$.

Let’s calculate eigenvalues for the following matrix:

$\begin{split} A=\begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}\end{split}$

#### Find eigenvalues¶

Looking at the above recipe, let’s solve the problem symbollically using sympy. First lets create a matrix $$B$$ such that:

$B = A-\lambda I$
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing()

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
<ipython-input-1-97c04e533e83> in <module>
----> 1 get_ipython().run_line_magic('matplotlib', 'inline')
2 import matplotlib.pylab as plt
3 import numpy as np
4 import sympy as sym
5 sym.init_printing()

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in run_line_magic(self, magic_name, line, _stack_depth)
2342                 kwargs['local_ns'] = self.get_local_scope(stack_depth)
2343             with self.builtin_trap:
-> 2344                 result = fn(*args, **kwargs)
2345             return result
2346

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/decorator.py in fun(*args, **kw)
230             if not kwsyntax:
231                 args, kw = fix(args, kw, sig)
--> 232             return caller(func, *(extras + args), **kw)
233     fun.__name__ = func.__name__
234     fun.__doc__ = func.__doc__

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magic.py in <lambda>(f, *a, **k)
185     # but it's overkill for just that one bit of state.
186     def magic_deco(arg):
--> 187         call = lambda f, *a, **k: f(*a, **k)
188
189         if callable(arg):

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magics/pylab.py in matplotlib(self, line)
97             print("Available matplotlib backends: %s" % backends_list)
98         else:
---> 99             gui, backend = self.shell.enable_matplotlib(args.gui.lower() if isinstance(args.gui, str) else args.gui)
100             self._show_matplotlib_backend(args.gui, backend)
101

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in enable_matplotlib(self, gui)
3511         """
3512         from IPython.core import pylabtools as pt
-> 3513         gui, backend = pt.find_gui_and_backend(gui, self.pylab_gui_select)
3514
3515         if gui != 'inline':

~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/pylabtools.py in find_gui_and_backend(gui, gui_select)
278     """
279
--> 280     import matplotlib
281
282     if gui and gui != 'auto':

ModuleNotFoundError: No module named 'matplotlib'

#Most sympy requires defeing the variables as "symbols"
#Once we do this we can use the variables in place of numbers
lam = sym.symbols('lambda')

A = sym.Matrix([[0, 0 ,-2], [1, 2, 1], [1, 0, 3]])
I = sym.eye(3)

B = A - lam*I

B


Now, per step 2, the determinate of $$B$$ must be zero. Note that sympy calculates the determinate symbollically as follows:

B.det()


Do This: Using the sympy.solve function on the determinate of $$B$$ to solve for lam ($$\lambda$$). Verify that the solution to the last question produces the same eigenvalues as above.

# Put your code to solve for det(B) = 0 here


Do This: First, let’s use the built in funciton eigenvals function in sympy to calculate the eigenvalues. Find out the meaning of the output.

# Put your code here


#### Find eigenvectors¶

Now we know the eigenvalues, we can substitue them back into the equation to find the eigenvectors.
We solve this symbollically using sympy. First let’s make a vector of our eigenvalues (from above):

eig = [1,2]


Now (per step 4 above) we need to solve the equation $$(A-\lambda I)x=0$$. One way to do this in sympy is as follows:

x1,x2,x3 = sym.symbols(['x_1','x_2','x_3'])

x = sym.Matrix([[x1],[x2],[x3]])
x

for lam in eig:
vec = sym.solve((A - lam*I)*x,x)
print(vec)


QUESTION: Explain your output here. (Hint, you can also try the rref to find the solutions)

Do This: Next, let’s use the eigenvects function in sympy to find three linear independent eigenvectors for the matrix $$A$$?

# Put your answer to the above question here


QUESTION: Compare this answer to the eigenvectors we calculated above. Does this answer make sense? What does the syntax tell us?

DO THIS: Find the eigenvalues and eigenvectors of the following matrix: $$$A2=\begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}$$$

QUESTION: What are the eigenvalues for the matrix $$A2$$?

QUESTION: What are the eigenvectors for the matrix $$A2$$?

## 2. Diagonalizable Matrix¶

In class we will be using matrix diagonalization to solve some problems.

Matrix $$A$$ is diagonalizable if there exists a diagonal matrix $$D$$ that is similar similar to $$A$$:

$D = C^{-1}AC$

If matrix $$A$$ has linearly independent eigenvectors ($$v_1, \ldots v_n$$) then $$A$$ is diagonalizable with the following solution:

$C = \left[ v_1^T, \ldots, v_n^T \right]$

In other words, each column of $$C$$ is a linearly independent eigenvector of $$A$$. The diagonal matrix $$D$$ is

$\begin{split} D = \left[ \begin{matrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n \end{matrix} \right] \end{split}$

In other-other words, $$D$$ consists of the corresponding eigenvalues.

%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)


DO THIS: Using numpy, Diagonalize (i.e. calculate C and D) the following matrix:

A = np.matrix([[5, -2, 2], [4, -3, 4], [4,-6,7]])
sym.Matrix(A)

# Put your answer here

from answercheck import checkanswer



DO THIS: Verify that $$A$$ is in fact Diagonalizable by calculating $$D2 = C^{-1}AC$$ and comparing it to your original $$D$$ using np.allclose.

#Put your verificaiton code here.

np.allclose(D,D2)


### Diagonalization of Symmetric Matrices¶

One special case is Symmetric Matrices. It can be shown that symmetric Matrices are Diagonalizable and the resulting eigenvectors are not only linearly independent but also orthogonal. Since this is true, the equation changes to:

$D = C^{T}AC$

QUESTION: Why do we care if $$C$$ is orthogonal? What advantages does the above equation give us?

## 3. Assignment wrap-up¶

Assignment-Specific QUESTION: Why do we care if $$C$$ is orthogonal? What advantages does the above equation give us?

QUESTION: Summarize what you did in this assignment.

QUESTION: What questions do you have, if any, about any of the topics discussed in this assignment after working through the jupyter notebook?

QUESTION: How well do you feel this assignment helped you to achieve a better understanding of the above mentioned topic(s)?

QUESTION: What was the most challenging part of this assignment for you?

QUESTION: What was the least challenging part of this assignment for you?

QUESTION: What kind of additional questions or support, if any, do you feel you need to have a better understanding of the content in this assignment?

Written by Dr. Dirk Colbry, Michigan State University 