Prerequisites of linear algebra for machine learning

Just about everyone has watched animated movies such as Frozen or Big Hero 6 or has at least heard about 3D computer games. It seems more fun to enjoy the movies and games rather than reading a Linear Algebra book. But it is because of linear algebra that we are able to watch a character move on the screen. Linear Algebra is the motivation of our new digital world.
Through this article, we will learn matrix arithmetic and learn how to use numpy to carry out these operations in python.

Why we need Linear Algebra for Machine Learning?

Machine Learning deals with the handling of enormous data sets. And an effective way to represent this data is in the form of 2D arrays or rectangular blocks in which each row represents a sample or a complete record and a column represents a feature or an attribute. It is natural to think of the array as a matrix and each column (attribute) as a vector.

Python and Linear Algebra

Numpy is a library used for scientific computing in Python. It provides multidimensional arrays and tools for working with these arrays.

Matrices

A matrix is a rectangular array of numbers (usually real) represented in forms of rows and columns. For example,

\(\begin{bmatrix}1 & 2 & 3\\100 & -3 & 1.15\end{bmatrix}\)

The dimension of a matrix is written as number of rows \(\times\) number of columns. So the dimension of above matrix is 2[math]\times[/math]3. A general m[math]\times[/math]n matrix is of the form

A = \(\left[\begin{array}{cccc}a_{11} & a_{12} & \dots & a_{1n}\\a_{21} & a_{22} & \dots & a_{2n}\\\dots&\dots&\dots&\dots\\a_{m1} & a_{m2} & \dots & a_{mn}\end{array}\right]=\left(a_{ij}\right)_{m\times n}\)

.array()

It helps define multidimensional arrays.

>>>import numpy as np
>>>A=np.array([[1,2,2],[3,2,1]])
>>>print A
 [[1 2 2]
  [3 2 1]]

matrix_name.shape

Returns (number of rows, number of columns)

>>>import numpy as np
>>>A=np.array([[1,2,2],[3,2,1]])
>>>print A.shape
(2,3)

Identity Matrix

An identity matrix, I of dimension n[math]\times[/math]n is the matrix in which all the elements on principal diagonal are 1 and the rest are 0.

.eye(n)

Returns n[math]\times[/math]n Identity Matrix

>>>import numpy as np
>>>A=np.eye(3)
[[1 0 0]
 [0 1 0]
 [0 0 1]]

Square Matrix

A matrix is called a square matrix if the number of rows is equal to the number of columns. An identity matrix is always a square matrix.

Matrix Operations

Matrices are mathematical objects on which we can perform multiple operations like addition, multiplication, etc.

• Addition

  Addition of matrices is only defined for matrices of the same dimension as addition is done entry wise, i.e., for two matrices

  A = \(\left(a_{ij}\right)_{m\times n}\) and B = \(\left(b_{ij}\right)_{m\times n} \)

  A+B = \(\left(a_{ij}+b_{ij}\right)_{m\times n}\)

  Suppose Amy, Sheldon, Leonard, Raj, and Alex have 3000 $ each to invest in Mutual Funds of Reliance and Airtel in September 2016.

   

  Name	Investment in Reliance MF	Investment in Airtel MF
  Amy	1500	1500
  Sheldon	2300	700
  Leonard	0	3000
  Raj	1000	2000
  Alex	1900	1100

  All of them also made investments in August 2016 as follows-

  Name	Investment in Reliance MF	Investment in Airtel MF
  Amy	1000	2000
  Sheldon	2100	900
  Leonard	3000	0
  Raj	1200	1800
  Alex	1400	1600

  Now if we have

  A =\(\begin{bmatrix}1500 & 1500\\2300&700\\0&3000\\1000&2000\\1900&1100\end{bmatrix}\) and B = \(\begin{bmatrix}1000 &2000\\2100&900\\3000&0\\1200&1800\\1400&1600\\\end{bmatrix}\)
  then A+B = C = \( \begin{bmatrix}2500 & 3500\\4400 & 1600\\3000&3000\\2200&3800\\3300&2700\end{bmatrix}\)

  Here C signifies the total investments made by Amy, Sheldon, Leonard, Raj and Alex in August and September. For instance, the total investment made by Leonard in Airtel Mutual Fund is \(c_{32}=3000\)$.

  .add(Matrix 1, Matrix 2)

  Returns Matrix 1 + Matrix 2

  >>>import numpy as np
  >>>A=np.array([[1,2,2],[2,3,4]])
  >>>B=np.array([[5,7,3],[3,-3,0]])
  >>>C=np.add(A,B)
  >>>print C
   [[6,9,6]
    [8,4,3]]    
  

  Some properties of addition of matrices are as follows-
  Associativity: \((A+B)+C = A+(B+C)\)
  Commutativity: \(A+B = B+A\)

• Transposition

  Transpose of a \(m\times n\) matrix, A is the matrix obtained by interchanging rows and columns of A. It is denoted by [math]A^\mathsf{T}[/math]. Clearly, [math]A^\mathsf{T}[/math] is a \(n\times m\) matrix.
  If A = \(\begin{bmatrix}1 & 2 & 3\\100 & -3 & 1.15\end{bmatrix}\) then

  [math] A^\mathsf{T}[/math]=\(\begin{bmatrix}1 & 100\\2 & -3 \\3& 1.15\end{bmatrix}\)

  .transpose(A)

  Returns [math] A^\mathsf{T}[/math]

  >>>import numpy as np
  >>>A=np.array([[1,2,2],[2,3,4]])
  >>>C=np.transpose(A)
  >>>print C
   [[1,2]
    [2,3]
    [2,4]]    
  

  Transposing will be useful in the multiplication of vectors.

• Multiplication

  Multiplication of any two arbitrary matrices is not possible. For two matrices to be compatible for multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix.
  That is, for two matrices A and B, AB exists only if the dimension of A is \(m\times n\) and the dimension of B is \(n\times q\). AB has dimension \(m\times q\).
  To multiply two matrices, we take the dot product of rows and columns, i.e., say, A =\(\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}\) and B =\(\begin{bmatrix}1&5&1\\3&2&0\\-1&1&3\end{bmatrix}\)

  This might look a little more abstract as compared to the addition of matrices. Let’s look at the intuition behind this.
  Consider the example we used above. Now, if Reliance MF gives a dividend of 10% in the month of September and Airtel had a loss and charges an extra 2%, what is the net gain/loss made by, say, Sheldon in the month of September?
  Net Gain/Loss= \(0.1\times2300\) – \(0.02\times700=216\)
  Let’s look at this in terms of matrices-

  New Reliance Mutual Fund   \(\begin{bmatrix}1.1 & 0\end{bmatrix}\)
  New Airtel Mutual Fund        \( \begin{bmatrix}0 & .98\end{bmatrix}\)
  Net Profit                               \( \begin{bmatrix}0.1 & -0.02\end{bmatrix}\)

  To calculate net gain by each investor for both the mutual funds we calculate

  \(\begin{bmatrix}1.1 & 0\\0&0.98\\0.1&-0.02\end{bmatrix}\begin{bmatrix}1500 & 1500\\2300&700\\0&3000\\1000&2000\\1900&1100\end{bmatrix}^\mathsf{T}=\begin{bmatrix}1.1 & 0\\0&0.98\\0.1&-0.02\end{bmatrix}\begin{bmatrix}1500&2300&0&1000&1900\\1500&700&3000&2000&1100\end{bmatrix}\)
  \(=\begin{bmatrix}1650 & 2530 & 0 & 1100 & 2090\\1470 & 686 & 2940 & 1960 & 1078\\120 & 216 & -60 & 60 & 168\end{bmatrix}\)

  So the above calculated matrix represents-

  	Amy	Sheldon	Leonard	Raj	Alex
  Current Value Of Reliance MF	1650	2530	0	1100	2090
  Current Value Of Airtel MF	1470	686	2940	1960	1078
  Net Gain	120	216	-60	60	168

  Following is a list of properties of multiplication of matrices-
  Associativity: \((A.B).C=A.(B.C)\)
  Non-commutativity: \(A.B \neq B.A\)
  Transposition rule: \((A.B)^T=B^{T}.A^{T}\)
  \(A.I=I.A=A\)

• Inversion

  We define the inverse of real number, [math]r[/math] as an other real number [math]r^{-1}[/math] such that [math]r.r^{-1}=1[/math]. For instance the inverse of 2 is \(\frac{1}{2}=0.5\) and that of 3 is \(\frac{1}{3}\). Not all real numbers have inverses like 0 because \(\frac{1}{0}\) is an undefined quantity.
  Similarly, we can define the inverse of a matrix A as \(A^{-1}\) such that \( A.A^{-1}=I\). Again, not all matrices have inverses. Rather only a category of square matrices is invertible. For instance the matrix [math]A=\begin{bmatrix}2&3\\4&5\end{bmatrix}[/math] has inverse[math]A^{-1}=\begin{bmatrix}-2.5 &1.5\\2&-1\end{bmatrix}[/math]

  .linalg.inv(Matrix name)

  It returns the inverse of a specified array.

  >>> import numpy as np
  >>> A=np.array([[2,3],[4,5]])
  >>> B=np.linalg.inv(A)
  >>> print B
   [[-2.5 1.0]
    [2.0 -1.0]]
  

Vectors

A vector is an n-tuple of values. Conventionally, it is represented as a \(n\times1\) matrix but can also be written as a \(1\times n\) matrix. The dimension of a n-tuple vector is n and n can be any positive integer from 1 to infinity. The general form of a n dimensional vector is

[math]v=\begin{pmatrix}v_1\\v_2\\\vdots \\v_n\end{pmatrix}[/math]

Vector arithmetic is similar to matrix arithmetic.

• Addition

  [math]v+w=\begin{pmatrix}v_1\\v_2\\\vdots \\v_n\end{pmatrix}+\begin{pmatrix}w_1\\w_2\\\vdots \\w_n\end{pmatrix}
  =\begin{pmatrix}v_1\mbox{ + }w_1\\v_2\mbox{ + }w_2\\\vdots \\v_n\mbox{ + }w_n\end{pmatrix}[/math]

• Multiplication

  Multiplication of two vectors is similar to the dot product of two vectors which returns a scalar.

  [math]v.w=vw^{T}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}\begin{pmatrix}w_1&w_2
  &\ldots&w_n\end{pmatrix}=\begin{pmatrix}v_1w_1+v_2w_2+\ldots+v_nw_n\end{pmatrix}[/math]

  .dot(v,w)

  Returns \(vw^{T}\)

  >>> import numpy as np
  >>> A=np.array([[1,2,3]])
  >>> B=np.array([[5,6,7]])
  >>> C=np.dot(A,B)
  >>> print C
   38.0
  

In the next article we will see the application of matrix arithmetic to principal component analysis which deals with identifying patterns in data.