Hi. I wanted to play around with linear algebra Python, and the Numpy package. In the past, I have used MATLAB for linear algebra work but nowadays MATLAB does not seem like a viable option with Python's Numpy package being around.

Sections

Some Vector Operations
Some Matrix Operations
References

Some Vector Operations

Creating Vectors

To start, here are two vectors a = (1, -3, 5) and b = (2, -1, 7).

>>> ## Vectors:
... 
... a = np.array([1, -3, 5])
... b = np.array([2, -1, 7])
... 
... print(a)
... print(b)
[ 1 -3  5]
[ 2 -1  7]

Vector Addition

With vector addition and vector subtraction, it operates element-wise.

>>> print(a + b) # Vector addition
... print(a - b) # Vector subtraction
[ 3 -4 12]
[-1 -2 -2]

Scalar Multiplication

Here is an example of scalar multiplication on a vector.

>>> print(2*a) # Scalar multiplication on vector
[ 2 -6 10]

Dot Products Of Vectors

Given two vectors (of equal size), one can compute the dot product of the two vectors.

>>> # Dot Product (Element wise multiplication and take sum of products):
... 
... print(np.dot(a, b)) # (1*2) + (-3)(-1) + 7*5 = 2 + 3 + 35 = 40
40

Cross Products Of Vectors

The cross product of two vectors is a (third) vector that is perpendicular/orthogonal to both vectors. (i.e. The z-axis is orthogonal to both the x-axis and the y-axis.)

Cross Product Image Source

>>> # Cross Product (A Vector That Is Perpendicular/Orthogonal To Vectors a and b):
... 
... cross_vect = np.cross(a, b)
... print(cross_vect)
[-16   3   5]

Using Dot Products To Check Orthogonality

To make sure that this (-16, 3, 5) vector is indeed orthogonal to the vectors a = (1, -3, 5) and b = (2, -1, 7) the dot product is used. If the dot product is zero between two vectors, then those two vectors are orthogonal to each other.

# Check orthogonality by seeing dot product is zero between cross product with a and b:
... 
... print(np.dot(cross_vect, a))
... 
... print(np.dot(cross_vect, b))
[-16   3   5]
0
0

Math Function Example

Here is an example with a math function. I have a list of x-values being 0, 1, 2 and 3. These x-values are inputted into the exponential function.

>>> # Function example:
... 
... x = np.array([0, 1, 2, 3])
... 
... print(np.exp(x))
[  1.           2.71828183   7.3890561   20.08553692]

Random Number Generation Example

Random numbers can be generated element-wise. In this example, I generate random numbers from a normal distribution.

>>> # Vector with random numbers from a normal distribution.
... 
... print(np.random.normal(size = (1, 4)))
[[-1.87577226 -0.77085276 -0.39107313  0.36635035]]

Some Matrix Operations

Special Matrices

To start this section, here are some special matrices such as the identity matrix, the matrix with zeroes and the matrix of ones.

>>> ## Matrices:
... 
... # 5 by 5 Identity Matrix
... 
... identity_five = np.identity(5)
... 
... print(identity_five)
[[ 1.  0.  0.  0.  0.]
 [ 0.  1.  0.  0.  0.]
 [ 0.  0.  1.  0.  0.]
 [ 0.  0.  0.  1.  0.]
 [ 0.  0.  0.  0.  1.]]

>>> # 3 by 3 Matrix of zeroes:
... 
... print(np.zeros( (3, 3)))
... 
[[ 0.  0.  0.]
 [ 0.  0.  0.]
 [ 0.  0.  0.]]

>>> # 3 by 3 Matrix of ones:
... 
... print(np.ones( (3, 3)))
[[ 1.  1.  1.]
 [ 1.  1.  1.]
 [ 1.  1.  1.]]

Creating A Matrix In Numpy

Creating a matrix in Numpy can be confusing. You may ask yourself, which one is a row and which one is a column. Matrix rows are represented by the lists inside the one list. In the example below, the first row has the elements 1, 5 and 2, the second row has 2, 3 and -1 and the third row has 0, 1 and - 5.

Matrix Image Source

>>> # Creating a matrix:
... 
... A = np.array([[1., 5., 2.], [2., 3.,-1.], [0., 1., -5.]])
... 
... print(A)
[[ 1.  5.  2.]
 [ 2.  3. -1.]
 [ 0.  1. -5.]]

Determinant Of A Matrix

Given a square matrix, the determinant can be determined with the det() function from the linear algebra package.

>>> # Matrix determinant:
... 
... det_A = np.linalg.det(A)
... 
... print(det_A)
... 
40.0

Diagonal Of A Matrix

The diagonal of a matrix consists of the numeric entries along the main diagonal. In matrix A, we have 1, 3 and -5.

>>> # Diagonal of Matrix:
... 
... print(np.diag(A))
[ 1.  3. -5.]

Trace Of A Matrix

The trace of a square matrix is the sum of the matrix's diagonal entries. From matrix A, the trace would be 1 + 3 - 5 = -1.

>>> # Trace Of A Matrix:
... 
... print(np.trace(A))
-1.0

Transpose Of A Matrix

The transpose of a matrix has the rows and the columns switched.

>>> # Transpose Of Matrix:
... 
... print(np.transpose(A)) # Transpose of matrix A
... 
[[ 1.  2.  0.]
 [ 5.  3.  1.]
 [ 2. -1. -5.]]

Inverse Of A Matrix

>>> # Inverse Of A Square Matrix:
... 
... print(np.linalg.inv(A))
[[-0.35   0.675 -0.275]
 [ 0.25  -0.125  0.125]
 [ 0.05  -0.025 -0.175]]

Matrix Multiplication

Matrix multiplication involves at least two matrices and operates differently than typical multiplication. In math notation, I have: