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Diffstat (limited to 'matrices/slides.org')
| -rw-r--r-- | matrices/slides.org | 93 |
1 files changed, 49 insertions, 44 deletions
diff --git a/matrices/slides.org b/matrices/slides.org index 4be93d2..b99bf7b 100644 --- a/matrices/slides.org +++ b/matrices/slides.org @@ -13,6 +13,7 @@ #+LaTeX_HEADER: \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet} #+LaTeX_HEADER: \usepackage{listings} +#+LaTeX_HEADER: \usepackage{amsmath} #+LaTeX_HEADER:\lstset{language=Python, basicstyle=\ttfamily\bfseries, #+LaTeX_HEADER: commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen}, @@ -63,37 +64,23 @@ - Getting started with Lists. - Getting started with Arrays. - Accessing parts of Arrays. -* Creating a matrix - - Creating a matrix using direct data - : In []: m1 = array([1, 2, 3, 4]) - - Creating a matrix using lists - : In []: l1 = [[1,2,3,4],[5,6,7,8]] - : In []: m2 = array(l1) * Exercise 1 - Create a (2, 4) matrix ~m3~ - : m3 = [[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 - + - Create a two dimensional matrix ~m3~ of order (2, 4) with + elements 5, 6, 7, 8, 9, 10, 11, 12. * 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~ + The following functions can also be used with matrices + - ~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 + - 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~ * Exercise 2 : Frobenius norm \& inverse - Find out the Frobenius norm of inverse of a ~4 X 4~ matrix. + - Find out the Frobenius norm of inverse of a ~4 X 4~ matrix. : The matrix is : m5 = arange(1,17).reshape(4,4) @@ -108,7 +95,7 @@ $||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}$ #+end_latex * Exercise 3 : Infinity norm - Find the infinity norm of the matrix ~im5~ + - Find the infinity norm of the matrix ~im5~ : - Infinity norm is defined as, #+begin_latex @@ -120,17 +107,11 @@ - Infinity norm : In []: norm(im5, ord=inf) * 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) + eigen values and eigen vectors + - eig() + + Only eigen values + - eigvals() * Singular Value Decomposition (~svd~) #+begin_latex $M = U \Sigma V^*$ @@ -153,15 +134,39 @@ - Create matrices using arrays. - Add and multiply the elements of matrix. - - Find out the inverse of a matrix,using the function ``inv()``. - - Use the function ``det()`` to find the determinant of a matrix. + - Find out the inverse of a matrix,using the function ``inv()''. + - Use the function ``det()'' to find the determinant of a matrix. - Calculate the norm of a matrix using the for loop and also using - the function ``norm()``. + the function ``norm()''. - Find out the eigen vectors and eigen values of a matrix, using - functions ``eig()`` and ``eigvals()``. + functions ``eig()'' and ``eigvals()''. - Calculate singular value decomposition(SVD) of a matrix using the - function ``svd()``. + function ``svd()''. +* Evaluation +1. A and B are two array objects. Element wise multiplication in + matrices are done by, + + - A * B + - multiply(A, B) + - dot(A, B) + - element\_multiply(A,B) + +2. ``eig(A)[ 1 ]'' and ``eigvals(A)'' are the same. + + - True + - False + +3. ``norm(A,ord='fro')'' is the same as ``norm(A)'' ? + + - True + - False +* Solutions +1. A * B + +2. False + +3. True * #+begin_latex \begin{block}{} |