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#+TITLE: Matrices

#+AUTHOR: FOSSEE

#+EMAIL:     

#+DATE:    

#+DESCRIPTION: 

#+KEYWORDS: 

#+LANGUAGE:  en

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* Outline

  - Creating Matrices

    - using direct data

    - converting a list

  - Matrix operations

  - Inverse of matrix

  - Determinant of matrix

  - Eigen values and Eigen vectors of matrices

  - Norm of matrix

  - Singular Value Decomposition of matrices

* Creating a matrix

  - Creating a matrix using direct data

  : In []: m1 = matrix([1, 2, 3, 4])

  - Creating a matrix using lists

  : In []: l1 = [[1,2,3,4],[5,6,7,8]]

  : In []: m2 = matrix(l1)

  - A matrix is basically an array

  : In []: m3 = array([[5,6,7,8],[9,10,11,12]])

* Matrix operations

  - Element-wise addition (both matrix should be of order ~mXn~)

    : In []: m3 + m2

  - Element-wise subtraction (both matrix should be of order ~mXn~)

    : In []: m3 - m2

* Matrix Multiplication

  - Matrix Multiplication

    : In []: m3 * m2

    : Out []: ValueError: objects are not aligned

  - Element-wise multiplication using ~multiply()~

    : multiply(m3, m2)

* Matrix Multiplication (cont'd)

  - Create two compatible matrices of order ~nXm~ and ~mXr~

    : In []: m1.shape

    - matrix m1 is of order ~1 X 4~

  - Creating another matrix of order ~4 X 2~

    : In []: m4 = matrix([[1,2],[3,4],[5,6],[7,8]])

  - Matrix multiplication

    : In []: m1 * m4

* Recall from ~array~

  - The functions 

    - ~identity(n)~ - 

      creates an identity matrix of order ~nXn~

    - ~zeros((m,n))~ - 

      creates a matrix of order ~mXn~ with 0's

    - ~zeros_like(A)~ - 

      creates a matrix with 0's similar to the shape of matrix ~A~

    - ~ones((m,n))~

      creates a matrix of order ~mXn~ with 1's

    - ~ones_like(A)~

      creates a matrix with 1's similar to the shape of matrix ~A~

  Can also be used with matrices

* More matrix operations

  Transpose of a matrix

  : In []: m4.T

* Exercise 1 : Frobenius norm \& inverse

  Find out the Frobenius norm of inverse of a ~4 X 4~ matrix.

  : 

  The matrix is

  : m5 = matrix(arange(1,17).reshape(4,4))

  - Inverse of A, 

    - 

     #+begin_latex

       $A^{-1} = inv(A)$

     #+end_latex

  - Frobenius norm is defined as,

    - 

      #+begin_latex

        $||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}$

      #+end_latex

* Exercise 2: Infinity norm

  Find the infinity norm of the matrix ~im5~

  : 

  - Infinity norm is defined as,

    #+begin_latex

       $max([\sum_{i} abs(a_{i})^2])$

    #+end_latex

* ~norm()~ method

  - Frobenius norm

    : In []: norm(im5)

  - Infinity norm

    : In []: norm(im5, ord=inf)

* Determinant

  Find out the determinant of the matrix m5

  : 

  - determinant can be found out using

    - ~det(A)~ - returns the determinant of matrix ~A~

* eigen values \& eigen vectors

  Find out the eigen values and eigen vectors of the matrix ~m5~.

  : 

  - eigen values and vectors can be found out using

    : In []: eig(m5)

    returns a tuple of /eigen values/ and /eigen vectors/

  - /eigen values/ in tuple

    - ~In []: eig(m5)[0]~

  - /eigen vectors/ in tuple

    - ~In []: eig(m5)[1]~

  - Computing /eigen values/ using ~eigvals()~

    : In []: eigvals(m5)

* Singular Value Decomposition (~svd~)

  #+begin_latex

    $M = U \Sigma V^*$

  #+end_latex

    - U, an ~mXm~ unitary matrix over K.

    - 

      #+begin_latex

        $\Sigma$

      #+end_latex

	, an ~mXn~ diagonal matrix with non-negative real numbers on diagonal.

    - 

      #+begin_latex

        $V^*$

      #+end_latex

	, an ~nXn~ unitary matrix over K, denotes the conjugate transpose of V.

  - SVD of matrix ~m5~ can be found out as,

    : In []: svd(m5)

* Summary

  - Matrices

    - creating matrices

  - Matrix operations

  - Inverse (~inv()~)

  - Determinant (~det()~)

  - Norm (~norm()~)

  - Eigen values \& vectors (~eig(), eigvals()~)

  - Singular Value Decomposition (~svd()~)

* Thank you!

#+begin_latex

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  This spoken tutorial has been produced by the

  \textcolor{blue}{FOSSEE} team, which is funded by the 

  \end{center}

  \begin{center}

    \textcolor{blue}{National Mission on Education through \\

      Information \& Communication Technology \\ 

      MHRD, Govt. of India}.

  \end{center}  

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#+end_latex