matrices/script.rst
author Anoop Jacob Thomas<anoop@fossee.in>
Sun, 07 Nov 2010 16:56:37 +0530
changeset 397 4aea9ed09983
parent 394 1a79f9ee7f5c
child 404 3117f104c577
permissions -rw-r--r--
made changes to matrix questions.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
319
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     1
.. Objectives
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     2
.. ----------
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     3
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     4
.. At the end of this tutorial, you will be able to 
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     5
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     6
.. 1. Create matrices using data.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     7
.. #. Create matrices from lists.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     8
.. #. Basic matrix operations.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
     9
.. #. Use ``inv()`` function to find inverse of a matrix.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    10
.. #. Use ``det()`` function to find determinant of a matrix.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    11
.. #. Use ``eig()`` and ``eigvals()`` functions to find eigen values
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    12
      and vectors
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    13
.. #. Use ``norm()`` function to find norm of a matrix.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    14
.. #. Use ``svd()`` function to find singular value decomposition of a
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    15
      matrix.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    16
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    17
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    18
.. Prerequisites
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    19
.. -------------
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    20
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    21
..   1. should have ``ipython`` and ``pylab`` installed. 
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    22
..   #. getting started with ``ipython``.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    23
..   #. getting started with lists.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    24
..   #. getting started with arrays.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    25
..   #. accessing part of arrays.
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    26
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    27
     
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    28
.. Author              : Anoop Jacob Thomas <anoop@fossee.in>
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    29
   Internal Reviewer   : 
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    30
   External Reviewer   :
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    31
   Checklist OK?       : <put date stamp here, if OK> [2010-10-05]
e8c02b3c51ac changed headers of scripts(objectives and prerequisites).
Anoop Jacob Thomas<anoop@fossee.in>
parents: 307
diff changeset
    32
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    33
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    34
========
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    35
Matrices
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    36
========
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    37
{{{ show the welcome slide }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    38
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    39
Welcome to the spoken tutorial on Matrices.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    40
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    41
{{{ switch to next slide, outline slide }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    42
307
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
    43
In this tutorial we will learn about matrices, creating matrices using
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
    44
direct data, by converting a list, matrix operations. Finding inverse
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
    45
of a matrix, determinant of a matrix, eigen values and eigen vectors
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
    46
of a matrix, norm and singular value decomposition of matrices.
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    47
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    48
{{{ creating a matrix }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    49
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    50
All matrix operations are done using arrays. Thus all the operations
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    51
on arrays are valid on matrices also. A matrix may be created as,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    52
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    53
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    54
    m1 = array([1,2,3,4])
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    55
340
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
    56
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
    57
.. #[Puneeth: don't use ``matrix``. Use ``array``. The whole script will
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
    58
.. have to be fixed.]
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
    59
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    60
Using the tuple ``m1.shape`` we can find out the shape or size of the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    61
matrix,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    62
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    63
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    64
    m1.shape
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    65
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    66
Since it is a one row four column matrix it returned a tuple, one by
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    67
four.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    68
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    69
A list can be converted to a matrix as follows,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    70
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    71
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    72
    l1 = [[1,2,3,4],[5,6,7,8]]
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    73
    m2 = array(l1)
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    74
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    75
{{{ switch to next slide, exercise 1}}}
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    76
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    77
Pause here and create a two dimensional matrix m3 of order 2 by 4 with
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    78
elements 5, 6, 7, 8, 9, 10, 11, 12.
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    79
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    80
{{{ switch to next slide, solution }}}
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    81
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
    82
m3 can be created as,
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    83
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    84
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    85
    m3 = array([[5,6,7,8],[9,10,11,12]])
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    86
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    87
{{{ switch to next slide, matrix operations }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    88
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    89
We can do matrix addition and subtraction as,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    90
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    91
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    92
    m3 + m2
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    93
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    94
does element by element addition, thus matrix addition.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    95
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    96
Similarly,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    97
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    98
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
    99
    m3 - m2
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   100
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   101
it does matrix subtraction, that is element by element
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   102
subtraction. Now let us try,
287
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   103
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   104
{{{ Switch to next slide, Matrix multiplication }}}
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   105
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   106
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   107
    m3 * m2
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   108
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   109
Note that in arrays ``m3 * m2`` does element wise multiplication and not
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   110
matrix multiplication,
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   111
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   112
And matrix multiplication in matrices are done using the function ``dot()``
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   113
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   114
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   115
    dot(m3, m2)
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   116
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   117
but due to size mismatch the multiplication could not be done and it
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   118
returned an error,
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   119
307
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   120
{{{ switch to next slide, Matrix multiplication (cont'd) }}}
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   121
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   122
Now let us see an example for matrix multiplication. For doing matrix
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   123
multiplication we need to have two matrices of the order n by m and m
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   124
by r and the resulting matrix will be of the order n by r. Thus let us
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   125
first create two matrices which are compatible for multiplication.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   126
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   127
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   128
    m1.shape
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   129
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   130
matrix m1 is of the shape one by four, let us create another one of
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   131
the order four by two,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   132
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   133
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   134
    m4 = array([[1,2],[3,4],[5,6],[7,8]])
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   135
    dot(m1, m4)
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   136
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   137
thus the function ``dot()`` can be used for matrix multiplication.
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   138
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   139
{{{ switch to next slide, recall from arrays }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   140
307
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   141
As we already saw in arrays, the functions ``identity()`` which
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   142
creates an identity matrix of the order n by n, ``zeros()`` which
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   143
creates a matrix of the order m by n with all zeros, ``zeros_like()``
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   144
which creates a matrix with zeros with the shape of the matrix passed,
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   145
``ones()`` which creates a matrix of order m by n with all ones,
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   146
``ones_like()`` which creates a matrix with ones with the shape of the
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   147
matrix passed. These functions can also be used with matrices.
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   148
307
1a73dddb1d05 added slides for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 287
diff changeset
   149
{{{ switch to next slide, more matrix operations }}}
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   150
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   151
To find out the transpose of a matrix we can do,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   152
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   153
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   154
    print m4
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   155
    m4.T
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   156
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   157
Matrix name dot capital T will give the transpose of a matrix
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   158
287
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   159
{{{ switch to next slide, Frobenius norm of inverse of matrix }}}
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   160
287
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   161
Now let us try to find out the Frobenius norm of inverse of a 4 by 4
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   162
matrix, the matrix being,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   163
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   164
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   165
    m5 = arange(1,17).reshape(4,4)
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   166
    print m5
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   167
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   168
The inverse of a matrix A, A raise to minus one is also called the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   169
reciprocal matrix such that A multiplied by A inverse will give 1. The
287
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   170
Frobenius norm of a matrix is defined as square root of sum of squares
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   171
of elements in the matrix. Pause here and try to solve the problem
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   172
yourself, the inverse of a matrix can be found using the function
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   173
``inv(A)``.
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   174
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   175
And here is the solution, first let us find the inverse of matrix m5.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   176
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   177
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   178
    im5 = inv(m5)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   179
287
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   180
And the Frobenius norm of the matrix ``im5`` can be found out as,
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   181
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   182
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   183
    sum = 0
394
1a79f9ee7f5c made changes to script, matrices, using array() instead of matrix() now.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 340
diff changeset
   184
    for each in im5.flatten():
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   185
        sum += each * each
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   186
    print sqrt(sum)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   187
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   188
{{{ switch to next slide, infinity norm }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   189
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   190
Now try to find out the infinity norm of the matrix im5. The infinity
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   191
norm of a matrix is defined as the maximum value of sum of the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   192
absolute of elements in each row. Pause here and try to solve the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   193
problem yourself.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   194
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   195
The solution for the problem is,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   196
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   197
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   198
    sum_rows = []
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   199
    for i in im5:
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   200
        sum_rows.append(abs(i).sum())
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   201
    print max(sum_rows)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   202
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   203
{{{ switch to slide the ``norm()`` method }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   204
287
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   205
Well! to find the Frobenius norm and Infinity norm we have an even easier
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   206
method, and let us see that now.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   207
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   208
The norm of a matrix can be found out using the method
287
d9507624eb8f added questions for matrices.
Anoop Jacob Thomas<anoop@fossee.in>
parents: 261
diff changeset
   209
``norm()``. Inorder to find out the Frobenius norm of the matrix im5,
261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   210
we do,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   211
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   212
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   213
    norm(im5)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   214
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   215
And to find out the Infinity norm of the matrix im5, we do,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   216
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   217
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   218
    norm(im5,ord=inf)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   219
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   220
This is easier when compared to the code we wrote. Do ``norm``
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   221
question mark to read up more about ord and the possible type of norms
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   222
the norm function produces.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   223
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   224
{{{ switch to next slide, determinant }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   225
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   226
Now let us find out the determinant of a the matrix m5. 
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   227
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   228
The determinant of a square matrix can be obtained using the function
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   229
``det()`` and the determinant of m5 can be found out as,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   230
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   231
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   232
    det(m5)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   233
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   234
{{{ switch to next slide, eigen vectors and eigen values }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   235
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   236
The eigen values and eigen vector of a square matrix can be computed
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   237
using the function ``eig()`` and ``eigvals()``.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   238
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   239
Let us find out the eigen values and eigen vectors of the matrix
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   240
m5. We can do it as,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   241
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   242
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   243
    eig(m5)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   244
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   245
Note that it returned a tuple of two matrices. The first element in
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   246
the tuple are the eigen values and the second element in the tuple are
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   247
the eigen vectors. Thus the eigen values are,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   248
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   249
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   250
    eig(m5)[0]
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   251
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   252
and the eigen vectors are,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   253
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   254
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   255
    eig(m5)[1]
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   256
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   257
The eigen values can also be computed using the function ``eigvals()`` as,
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   258
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   259
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   260
    eigvals(m5)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   261
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   262
{{{ switch to next slide, singular value decomposition }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   263
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   264
Now let us learn how to do the singular value decomposition or S V D
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   265
of a matrix.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   266
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   267
Suppose M is an m×n matrix whose entries come from the field K, which
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   268
is either the field of real numbers or the field of complex
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   269
numbers. Then there exists a factorization of the form
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   270
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   271
    M = U\Sigma V star
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   272
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   273
where U is an (m by m) unitary matrix over K, the matrix \Sigma is an
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   274
(m by n) diagonal matrix with nonnegative real numbers on the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   275
diagonal, and V*, an (n by n) unitary matrix over K, denotes the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   276
conjugate transpose of V. Such a factorization is called the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   277
singular-value decomposition of M.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   278
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   279
The SVD of matrix m5 can be found as
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   280
::
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   281
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   282
    svd(m5)
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   283
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   284
Notice that it returned a tuple of 3 elements. The first one U the
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   285
next one Sigma and the third one V star.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   286
    
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   287
{{{ switch to next slide, recap slide }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   288
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   289
So this brings us to the end of this tutorial. In this tutorial, we
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   290
learned about matrices, creating matrices, matrix operations, inverse
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   291
of matrices, determinant, norm, eigen values and vectors and singular
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   292
value decomposition of matrices.
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   293
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   294
{{{ switch to next slide, thank you }}}
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   295
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   296
Thank you!
c7f0069d698a added base scripts and questions except for matrices and other-type-of-plots. previous commit only removed unwanted files.
anoop
parents:
diff changeset
   297
340
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
   298
.. 
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
   299
   Local Variables:
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
   300
   mode: rst
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
   301
   indent-tabs-mode: nil
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
   302
   sentence-end-double-space: nil
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
   303
   fill-column: 75
3951809e75bd Changes suggested for Matrices LO.
Puneeth Chaganti <punchagan@fossee.in>
parents: 319
diff changeset
   304
   End: