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

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• Some Vector Operations
• Some Matrix Operations
• References

Some Vector Operations

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Creating Vectors

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

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

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Vector Addition

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

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

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Scalar Multiplication

Here is an example of scalar multiplication on a vector.

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

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Dot Products Of Vectors

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

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>>> # 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

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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

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>>> # 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]

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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.

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# 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

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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.

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

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Random Number Generation Example

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

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

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Some Matrix Operations

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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.

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>>> ## 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.]]

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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.]]

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Determinant Of A Matrix

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

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

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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.

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

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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.

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

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Transpose Of A Matrix

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

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

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Inverse Of A Matrix

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>>> # 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]]

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Matrix Multiplication

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

Matrix Image Source

>>> # Matrix multiplication:
... 
... B = np.array([[1, 3], [1, 2]]) # 2 by 2 matrix
... 
... D = np.array([[2], [-1]]) # 2 by 1 matrix
... 
... print(np.matmul(B, D))
[[-1]
 [ 0]]

Solving Linear Systems

Linear algebra is useful when it comes to solving linear systems. In this example, we seek values of x1 and x2 such that they satisfy the equations.

QuickLaTeX Image Source

The above system of equation can be expressed in linear algebra form like this:

QuickLaTeX Image Source

The linear system can be solved in Python's Numpy with the solve() function.

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>>> # Solving a linear system example Bx = c.
... 
... B = np.array([[1, 3], [1, 2]])
... 
... c = np.array([[2], [0]])
... 
... print(np.linalg.solve(B, c))
[[-4.]
 [ 2.]]

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The solutions would be x1 = -4 and x2 = 2. (since -4 + 6 = 2 and -4 + 4 = 0).

References

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• Python For Data Analysis Book (O'Reilly Media)
• https://docs.scipy.org/doc/numpy-dev/user/quickstart.html
• https://docs.scipy.org/doc/numpy/reference/routines.linalg.html
• Other Scipy documentation.