Hello friends and welcome to this tutorial on Matrices.

In python all matrix operations are done using arrays.

We have already seen in previous session that how arrays are better suited for certain mathematical operations. In this session we are going to cover more details on using Arrays as matrices, such as, how to create them, how to initialize them, how to manipulate and use them for solving given problem.

First things first, let's start by creating a normal array, type:

a equal to array([5, 8, 10, 13])

let's check the value of a by typing

a

Here a is single dimension array, that is it has only one row. Let's now look at creating multi-dimensional arrays by

c = array([[11,12,13], [21,22,23], [31,32,33]])

both c and a are arrays but with different dimensions or shape

we can check shape of arrays by using shape attribute of arrays.

a.shape

c.shape

A few other handy array initialization methods are also available to make life easier.

say we want to create an array of size 3x4 with all the elements initialized to 1, we use

b = ones((3, 4))

b

d = ones_like(c)

i = identity(3)

i

i = identity(5)

i

----------------

c

to access the element 23 we type

c[1][2]

c[1, 2]

here ',' is used as separator for row and column value. Similarly any value from the array can be accessed.

to access particular row completely we simply skip the column value

c[1]

c[1, 1] = -22

c

c[1] = 0

c

Accessing a row is straight forward we skip column part

c[:,2]

here the ':' part mentioned for row value symbolises entire row.

the c[1] we were using earlier can also be written as c[1,:]

':' actually takes two value. for any row or column we can mention

start:end values, and rows/columns starting for 'start' till 'end' will be returned. Lets try some examples for better understanding

c[0:2,:]

c[1:3,:] 

gives second and third row.

similarly we can try this on columns also:

c[:, 0:2] gives us first two column

This whole concept of accessing chunks of arrays is known as 'slicing'

There is one more interesting and handy feature of slicing. We saw earlier that how only ':' means entire row or column.

It actually means if we don't specify start and end part of slice default is from zero to end.

so

c[:, :2]

c[1:, :2]

we add one more ':' to row or column part to specify a step size.

c[:, ::2]

similarly 

Lets us try to use these concepts of slicing and striding for doing some basic image manipulation

a = imread('lena.png')

a is a numpy array with the 'RGB' values of each pixel

a.shape

its a 512x512x3 array.

to view the image write

imshow(a)

imshow(a[:255,:255]) (half of 512 is 256)

imshow(a)

imshow(a[200:400, 200:400])

imshow(a[::2, ::2])

note now the size of image is just 256x256 and still quality of image is not much compromised.

-------------------------

Till now we have covered initializing and accessing elements of arrays. Now we shall concentrate on functions available for arrays. We start this by creating 4x4 array by

a = array([[ 1, 1, 2, -1],[ 2, 5, -1, -9], [ 2, 1, -1, 3], [ 1, -3, 2, 7]])

a

To get transpose of this matrix write

a.T

sum(a)

b = array([[3,2,-1,5], [2,-2,4,9], [-1,0.5,-1,-7], [9,-5,7,3]])

a + b

lets try multiplication now, 

To get matrix product of a and b we use

dot(a, b)

inv(a)

det(a) returns determinant of matrix a

e = array([[3,2,4],[2,0,2],[4,2,3]])

and then to evaluate eigenvalues of array

it returns both eigen values and eigen vector of given matrix

to get only eigen values use

This brings us to end of this session. We have covered Matrices

Initialization

Slicing

Striding

A bit of image processing

Functions available for arrays

Thank you

----------------

We have seen 

    Welcome to the Tutorial on arrays. 

    As mentioned in [the previous tutorial] arrays are much faster and

    more efficient. In this tutorial we shall look at creating arrays,

    accessing elements and changing them. 

    ---

    Let's start with creating simple arrays. We've already seen how to

    convert lists to arrays. Inputting a new array is similar to that. 

    In []: 

    Type /a/, to see what it is. 

    In []: a

    We enter a multi-dimensional array this way -

    In []: c = array([[11,12,13],

                     [21,22,23],

                      [31,32,33]])

    To see what c is, we just type c in the prompt. 

    In []: c

    To see the dimensions of the array c, we use c.shape

    In []: c.shape 

    Now let us look at some special methods of creating an

    array. There are various functions that allow us to create special

    arrays. 

    The first one we shall look at is, /arange/. /arange/ is similar to

    the range command, except that it returns an array and accepts

    float arguments. 

    In []: a = arange(10)

    In []: a

    This is the array we just created. 

    In []: a.shape

    Note that /a/ is one dimensional and has 10 elements, as expected. 

    We could also use a.shape to change the shape of the array a. 

    In []: a.shape = 2,5

    Note that the total size of new array must be unchanged. 

    We type a, to see what it looks like

    In []: a

    ones command can be used to get an array with all the entries as

    ones. We pass it the shape of the array that we require. 

    In []: 

    Look at b, by printing it out. 

    In []: b 

    To create an array with all entries as ones, with it's shape

    similar to an already existing array, we use the ones_like

    command.  

    In []: b = ones_like(a)

    zeros and zeros_like are similar commands that can give you arrays

    with all zeros. empty and empty_like give you empty arrays (arrays

    with no initialization done.)

    In []: b = zeros((3, 4))

    In []: b = zeros_like(a)

    The identity command can be used to obtain a square array with

    ones on the main diagonal. 

    In []: identity(3)

    To obtain a 2-D array, that is not necessarily square, eye command

    can be used. Look at the documentation of eye (using eye?) for

    more info. 

    ---

    Now that we have learnt how to create arrays, let's move on to

    accessing elements and changing them. 

    Let's work with the c, array which we had already created. 

    In []: c 

    Let's say we want to access the element 23 in c, we say

    In []: c[1][2]

    Note that this is similar to accessing an element inside a list of

    lists. Also, note that counting again starts from 0. 

    But arrays provide a more convenient way to access the elements. 

    In []: c[1, 2]

    Now, we can also change the element using a simple assignment. 

    In []: c[1, 2] = -23

    Let's look at accessing more than one elements at a time. We begin

    with accessing rows. 

    In []: c[1] gives us the second row. (counting starts from 0)

    To get a column, we use a syntax that is similar to the one used

    to access a single element. 

    In []: c[:,1], gives us the first column. 

    The colon specifies that we wish to obtain all elements in that

    dimension from the array.  

    So, we could use a more explicit way to access the second row of

    the array. 

    In []: c[1,:]

    The colon can be used to access specific portions of the array,

    similar to the way we do with lists. 

    In []: c[1,1:3]

    Observe that we get the second and third columns from the second

    row. As with lists, the number after the colon is excluded when

    slicing a portion of the array. 

    In []: c[1:3,1]

    Now, we get the second and third rows from the first column. 

    In []: c[1:3,1:3]

    We get the second and third rows and the second and third

    columns. 

    The numbers before and after the colons are optional. If the

    number before the colon is omitted, it is assumed to be zero by

    default. If the element after the colon is omitted, it is assumed

    to be until the end. 

    In []: c[1:, 1:]

    This is essentially similar to the previous example. We are using

    the default value i.e, the end, instead of specifying 3,

    explicitly. 

    In []: c[:2, :2]

    We have omitted specifying the zero before the colon, explicitly. 

    --- 

    You may have observed the similarity of the semi-colon notation to

    the notation used in lists. As expected, the semi-colon notation

    also provides a way to specify a jump. This {concept/idea} is

    termed as Striding. 

    To get every alternate row of c, starting from the first one, we say

    In []: c[::2,:]

    To get every alternate row of c, starting from the second one, we

    say 

    In []: c[1::2,:]

    In []: c[:,::2]

    In []: c[::2,::2]

    ---

    We come to the end of this tutorial on arrays. In this tutorial,

    you've learnt how to create arrays and access, change elements. 

    Thank you. 

Hello friends and welcome to the second tutorial in the series of spoken tutorials on Python for Scientific computing. 

In the previous tutorial we learnt about arrays and we told you that numpy arrays are faster and more efficient . In this tutorial we shall look at creating arrays, accessing elements and changing them. 

Let's start with creating simple arrays. We've already seen how to convert lists to arrays. Inputting a new array is similarto that. 

On your Ipython terminal type a = array open parenthesis and then open square brackets 5,8,10,13 close square brackets and close parenthesis . This create an array a . You can see what a is by typing a on the terminal .

The first one is the command arange which is similar to range except that it returns an array.

Ones can be use to get all entries as ones . We can pass it the shape of the array as required .

type b=ones open parenthesis , another open parenthesis , 3,4 , close second parenthesis and close first parenthesis . Look at b , by printing it out .

To create an array with all entries as ones, with it's shape similar to an already existing array, we use the ones_like

command.  type b= ones_like in parenthesis a .