PPNM Ch. 14.1 (not all of this sub-chapter is covered in this lab)
Brilliant overview of matrices and matrix algebra
Brilliant overview of Linear Algebra
Optional: EMEA Ch. 12.1-12.7
Understand what a matrix and vector are
Understand basic matrix operations: equality, addition and multiplication, and transpose
Understand how to write a system of equations in matrix form
Know the form and definition of the identity matrix
Matrix multiplication
Transpose of a matrix
Basics of working with vectors and matrices in Python
In previous economics courses, you have met systems of equations: For example, price and quantity variables in both supply and demand equations that must be solved simultaneously. You likely have solved these in a somewhat ad-hoc way - manipulating one variable to be written as a function of another, and then substituting in order to eliminate a variable from an equation:
(ps. As it turns out, with the coefficients I chose for this example, you are not able to get a unique solution.)
Write down the system of equations for the case with (n=4) variables and (m=3) equations. and $a_{ij} = i+2j+(-1)^i$ for i=1,2,3 and j=1,2,3,4, whereas $b_i = 2^i$ for i=1,2,3.
(EMEA exercise 12.2.3)
One of our main goals in this lab and the next will be to systemize the solving of systems of equations, starting by writing the equations in terms of matrices and vectors. But first, we need to understand what matrices and vectors are:
Here is the Khan Academy introduction to matrices.
Write the coefficient matrix of the system of equations in the above exercise (12.2.3)
If we have 1 or more numbers, we can do a host of operations on these numbers: basically standard actions that manipulate these numbers with consistent results. For example, adding: we know that 2+2=4, and adding any other two numbers follow the same rules.
We have standard operations for matrices as well - sometimes that act similar to operations on numbers (called scalars in linear algebra world), but which sometimes are quite a bit different.
We start easy: Mattrix addition and scalar multiplication
Here is the Khan Academy video on matrix addition and subtraction.
Given a matrix $\mathbf{A} = \begin{bmatrix} 2,4 \\ -4, -2 \end{bmatrix}$ and $\mathbf{B} = \begin{bmatrix} -1,-2 \\ 1, 2 \end{bmatrix}$
Again, given a matrix $\mathbf{A} = \begin{bmatrix} 2,4 \\ -4, -2 \end{bmatrix}$ and $\mathbf{B} = \begin{bmatrix} -1,-2 \\ 1, 2 \end{bmatrix}$
Matrix multiply $\mathbf{AB}$
Matrix multiply $\mathbf{BA}$
Are the two the same?
As we stated in the beginning, one of the main goals of this lecture was to be able to write our system of equations in matrix and vector form. Now we are ready to do that:
Write the system of equations from the exercise above (12.2.3) in matrix form by creating matrix $\mathbf{A}$ and vectors $B$ and $x$.
As we saw in Video 1.4, matrix multiplication is not entirely analgous to multiplication of scalars. In this video, we talk a bit more about what we can and can not assume when we do matrix multiplication.
Verify the distributive law $\mathbf{A(B+C)} = \mathbf{AB + AC}$ where,
$$A=\begin{bmatrix} 1,2\\3,4\end{bmatrix}$$$$B=\begin{bmatrix} 2,-1, 1,0 \\3,-1,2,1\end{bmatrix}$$$$C=\begin{bmatrix} -1,1, 1,2 \\-2,2,0,-1\end{bmatrix}$$(EMEA Exercise 12.6.1)
Compute the following matrix product:
$$\begin{bmatrix}1,2,-3 \end{bmatrix} \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \end{bmatrix}$$In the previous videos, we have seen how the dimensions and ordering of matrices can be critically important in doing matrix calculations. Often we are interested in being able to transform the dimensions of our matrix, without changing the values in the matrix or changing the ordering of those values. One of our tools for doing this is the transpose:
Here is the Khan academy video on transpose.
Find the transposes of the three matrices:
$$\mathbf{A} = \begin{bmatrix}3,5,8,3 \\ -1,2,6,2 \end{bmatrix}$$$$\mathbf{B} = \begin{bmatrix}0\\1\\-1\\2 \end{bmatrix}$$$$\mathbf{C} = \begin{bmatrix} 1,5,0,-1 \end{bmatrix}$$We will be relying heavily on the np.array() data type
import numpy as np
NP arrays are fairly general and can contain many different sizes and dimensions of numbers
We can define both a row vector and a column vector
vector_row = np.array([[1, -5, 3, 2, 4]])
vector_column = np.array([[1],
[2],
[3],
[4]])
print(vector_row.shape)
print(vector_column.shape)
(1, 5) (4, 1)
Notice above that the inner brackets ([]) define the rows.
We could also define an array with 5 numbers:
myArray = np.array([1,-5,3,2,4])
myArray.shape
(5,)
But this array is just a collection of numbers, not a row- or column vector. A vector is defined a collection of numbers with a defined shape.
We can also calculate linear combinations of vectors:
x=np.array([[1,2,5]])
z=np.array([[2,3,7]])
y = 3*x + .5*z
print(y)
[[ 4. 7.5 18.5]]
We can also use numpy to create matrices. Really, there is no difference between vectors and matrices other than dimension: A vector is simply a 1xN or Nx1 matrix, and so it is in Python as well.
Below we create two matrices: P (3x2) and Q (2x4)
P = np.array([[1, 7], [2, 3], [5, 0]])
Q = np.array([[2, 6, 3, 1], [1, 2, 3, 4]])
print(P)
print(Q)
[[1 7] [2 3] [5 0]] [[2 6 3 1] [1 2 3 4]]
Again, notice the use of inner brackets to define rows.
To do matrix multiplication, we use the dot function (matrix multiplication is sometimes called the dot production. P is 3x2 and Q is 2x4, so we can calculate PQ, which should become a 3x4 matrix
np.dot(P, Q)
array([[ 9, 20, 24, 29],
[ 7, 18, 15, 14],
[10, 30, 15, 5]])
Increasingly, you will see the "@" sign used for matrix multiplication, this works exactly the same, and perhaps is even a bit easier to read:
P @ Q
array([[ 9, 20, 24, 29],
[ 7, 18, 15, 14],
[10, 30, 15, 5]])
But if we try QP:
np.dot(Q,P)
--------------------------------------------------------------------------- ValueError Traceback (most recent call last) /var/folders/fs/kr83xdyx2jq_8gww_xfmtxfr0000gn/T/ipykernel_56884/4123817340.py in <module> ----> 1 np.dot(Q,P) <__array_function__ internals> in dot(*args, **kwargs) ValueError: shapes (2,4) and (3,2) not aligned: 4 (dim 1) != 3 (dim 0)
Order matters!
We can automatically generate an identity matrix in python with a pun-y command "eye"
I = np.eye(3)
print(I)
[[1. 0. 0.] [0. 1. 0.] [0. 0. 1.]]
we can use the random functions in numpy to populate a matrix with random numbers. Here we generate random integers between 0 and 10 in a 3x4 matrix:
M = np.random.randint(10, size=(3, 4))
M
array([[7, 3, 1, 7],
[1, 2, 1, 7],
[7, 8, 8, 2]])
(Your results will be different, of course)
When we compute the IM we get our expected result
np.dot(I,M)
array([[7., 3., 1., 7.],
[1., 2., 1., 7.],
[7., 8., 8., 2.]])
Given the matrices:
$$\mathbf{A} = \begin{bmatrix} 2,0\\-1,1 \end{bmatrix}$$$$\mathbf{B} = \begin{bmatrix} -1,2\\1,-1 \end{bmatrix}$$$$\mathbf{C} = \begin{bmatrix}2,3\\1,4 \end{bmatrix}$$$$\mathbf{D} = \begin{bmatrix}1,1,1\\ 1,3,4\end{bmatrix}$$Calculate (where possible). If not possible, explain why
a.) $\mathbf{A-B}$
b.) $\mathbf{A+B-2C}$
c.) $\mathbf{AB}$
d.) $\mathbf{C(AB)}$
e.) $\mathbf{AD}$
f.) $\mathbf{DC}$
g.) $\mathbf{2A-3B}$
h.) $\mathbf{(A-B)´}$
i.) $\mathbf{(C´A´)B´}$
j.) $\mathbf{C´(A´B´)}$
j.) $\mathbf{D´D´}$
k.) $\mathbf{D´D}$
(EMEA review exercise 12.2)
(EMEA review exercise 12.3)
Find the matrices $\mathbf{A+B}$, $\mathbf{AB}$, $\mathbf{BA}$ and $\mathbf{A(BC)}$ where
$$\mathbf{A} = \begin{bmatrix} 0,1,-2 \\ 3,4,5 \\ -6,7,15 \end{bmatrix}$$$$\mathbf{B} = \begin{bmatrix} 0,-5,3 \\ 5,2, -1 \\-4,2,0 \end{bmatrix}$$$$\mathbf{C} = \begin{bmatrix} 6,-2,-3 \\2,0,1 \\0,5,7 \end{bmatrix}$$(EMEA review exercise 12.4)
Defining vectors:
$$x=[3,5,7]$$$$y=[2,4,8]$$a.) Create these vectors as np.arrays and compute the following linear combination: $.5x + 2y$
b.) Find the inner product of x and y, ($x \cdot y$)
Consider the two matrices
$$\mathbf{A_{3 \times 4}} = \begin{bmatrix} 2,4,6, 8 \\ 1,-2,4, -8 \\-3, 4, 5, -5 \end{bmatrix}$$$$\mathbf{B_{3 \times 4}} = \begin{bmatrix} 4,3,-1, 1 \\ -8,6,-4,2 \\1,2,2,1 \end{bmatrix}$$c.) Create matrices $\mathbf{A}$ and $\mathbf{B}$ as np.arrays. Confirm the shape of the matrices are 3x4. Calculate $2\mathbf{A}$
d.) Calculate $2\mathbf{A}+.5\mathbf{B}$
e.) Is it possible to compute the matrix multiplication $\mathbf{AB}$. What happens if you try?
f.) Instead, matrix multiply $\mathbf{A^TB}$ where $T$ represents the transpose (Hint: if you have an np.array, x, you can create the transpose by x.T
g.) Fill a 5x3 matrix, $\mathbf{R}$, with random integers. What shape of an identity matrix, $\mathbf{I}$ do we need so that $\mathbf{IM=M}$. Demonstrate computationally.
Consider 3 countries A, B, and C. Country A has 3 airports, country B has 4 airport, and country C has 5 airports. There are international flights between country A and B and between B and C, but no direct flights between A and C. The matrices below represent the number of direct flights between airports in the different countries.
$$\mathbf{AB_{3 \times 4}} = \begin{bmatrix} 2&4&0& 1 \\ 1&0&4,& 0 \\0& 2& 7& 3 \end{bmatrix} $$$$\mathbf{BC_{4 \times 5}} = \begin{bmatrix} 4&3&0& 1& 3 \\ 0&6&0&2& 0 \\2&5&1&0&3 \\1&7&0&2&3 \end{bmatrix} $$Do a calculation (in python) of these matrices to give the total number connections between countries A and C. (Hint, you wish to end up with a 3x5 matrix). Explain the logic behind this calculation.