

In [1]:

    
from sympy import *
init_printing(use_latex='mathjax')
x, y, z = symbols('x,y,z')
r, theta = symbols('r,theta', positive=True)

Matrices

The SymPy Matrix object helps us with small problems in linear algebra.



In [2]:

    
rot = Matrix([[r*cos(theta), -r*sin(theta)],
              [r*sin(theta),  r*cos(theta)]])
rot









    Out[2]:





$$\left[\begin{matrix}r \cos{\left (\theta \right )} & - r \sin{\left (\theta \right )}\\r \sin{\left (\theta \right )} & r \cos{\left (\theta \right )}\end{matrix}\right]$$

Standard methods



In [4]:

    
rot.det()









    Out[4]:





$$r^{2} \sin^{2}{\left (\theta \right )} + r^{2} \cos^{2}{\left (\theta \right )}$$



In [5]:

    
rot.inv()









    Out[5]:





$$\left[\begin{matrix}- \frac{\sin^{2}{\left (\theta \right )}}{r \cos{\left (\theta \right )}} + \frac{1}{r \cos{\left (\theta \right )}} & \frac{1}{r} \sin{\left (\theta \right )}\\- \frac{1}{r} \sin{\left (\theta \right )} & \frac{1}{r} \cos{\left (\theta \right )}\end{matrix}\right]$$



In [6]:

    
rot.singular_values()









    Out[6]:





$$\left [ r, \quad r\right ]$$

Exercise

Find the inverse of the following Matrix:

$$ \left[\begin{matrix}1 & x\\y & 1\end{matrix}\right] $$



In [7]:

    
# Create a matrix and use the `inv` method to find the inverse

Operators

The standard SymPy operators work on matrices.



In [8]:

    
rot * 2









    Out[8]:





$$\left[\begin{matrix}2 r \cos{\left (\theta \right )} & - 2 r \sin{\left (\theta \right )}\\2 r \sin{\left (\theta \right )} & 2 r \cos{\left (\theta \right )}\end{matrix}\right]$$



In [9]:

    
rot * rot









    Out[9]:





$$\left[\begin{matrix}- r^{2} \sin^{2}{\left (\theta \right )} + r^{2} \cos^{2}{\left (\theta \right )} & - 2 r^{2} \sin{\left (\theta \right )} \cos{\left (\theta \right )}\\2 r^{2} \sin{\left (\theta \right )} \cos{\left (\theta \right )} & - r^{2} \sin^{2}{\left (\theta \right )} + r^{2} \cos^{2}{\left (\theta \right )}\end{matrix}\right]$$



In [10]:

    
v = Matrix([[x], [y]])
v









    Out[10]:





$$\left[\begin{matrix}x\\y\end{matrix}\right]$$



In [11]:

    
rot * v









    Out[11]:





$$\left[\begin{matrix}r x \cos{\left (\theta \right )} - r y \sin{\left (\theta \right )}\\r x \sin{\left (\theta \right )} + r y \cos{\left (\theta \right )}\end{matrix}\right]$$

Exercise

In the last exercise you found the inverse of the following matrix



In [12]:

    
M = Matrix([[1, x], [y, 1]])
M









    Out[12]:





$$\left[\begin{matrix}1 & x\\y & 1\end{matrix}\right]$$



In [13]:

    
M.inv()









    Out[13]:





$$\left[\begin{matrix}\frac{x y}{- x y + 1} + 1 & - \frac{x}{- x y + 1}\\- \frac{y}{- x y + 1} & \frac{1}{- x y + 1}\end{matrix}\right]$$

Now verify that this is the true inverse by multiplying the matrix times its inverse. Do you get the identity matrix back?



In [14]:

    
# Multiply `M` by its inverse.  Do you get back the identity matrix?

Exercise

What are the eigenvectors and eigenvalues of M?



In [15]:

    
# Find the methods to compute eigenvectors and eigenvalues. Use these methods on `M`

NumPy-like Item access



In [14]:

    
rot[0, 0]









    Out[14]:





$$r \cos{\left (\theta \right )}$$



In [15]:

    
rot[:, 0]









    Out[15]:





$$\left[\begin{matrix}r \cos{\left (\theta \right )}\\r \sin{\left (\theta \right )}\end{matrix}\right]$$



In [17]:

    
rot[1, :]









    Out[17]:





$$\left[\begin{matrix}r \sin{\left (\theta \right )} & r \cos{\left (\theta \right )}\end{matrix}\right]$$

Mutation

We can change elements in the matrix.



In [18]:

    
rot[0, 0] += 1
rot









    Out[18]:





$$\left[\begin{matrix}r \cos{\left (\theta \right )} + 1 & - r \sin{\left (\theta \right )}\\r \sin{\left (\theta \right )} & r \cos{\left (\theta \right )}\end{matrix}\right]$$



In [19]:

    
simplify(rot.det())









    Out[19]:





$$r \left(r + \cos{\left (\theta \right )}\right)$$



In [20]:

    
rot.singular_values()









    Out[20]:





$$\left [ \sqrt{r^{2} + r \cos{\left (\theta \right )} + \frac{1}{2} \sqrt{4 r^{2} + 4 r \cos{\left (\theta \right )} + 1} + \frac{1}{2}}, \quad \sqrt{r^{2} + r \cos{\left (\theta \right )} - \frac{1}{2} \sqrt{4 r^{2} + 4 r \cos{\left (\theta \right )} + 1} + \frac{1}{2}}\right ]$$

Exercise

Play around with your matrix M, manipulating elements in a NumPy like way. Then try the various methods that we've talked about (or others). See what sort of answers you get.



In [21]:

    
# Play with matrices