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{| border=1
 
{| border=1
−
!Visual Cue
+
|'''Time'''
−
!Narration
+
|'''Narration'''
 +
 
 
|-
 
|-
−
| 0:01
+
| 00:01
 
| Hello friends and welcome to the tutorial on 'Matrices'.
 
| Hello friends and welcome to the tutorial on 'Matrices'.
 
 
 
|-
 
|-
−
| 0:05
+
| 00:05
 
| At the end of this tutorial, you will be able to,
 
| At the end of this tutorial, you will be able to,
−
 
+
Create matrices using data.
−
# Create matrices using data.
+
Create matrices from lists.
−
# Create matrices from lists.
+
Do basic matrix operations like addition,multiplication.
−
# Do basic matrix operations like addition,multiplication.
+
Perform operations to find out the -- - inverse of a matrix. - determinant of a matrix. - eigen values and eigen vectors of a matrix. - norm of a matrix. - singular value decomposition of a matrix.
−
# Perform operations to find out the -- - inverse of a matrix. - determinant of a matrix. - eigen values and eigen vectors of a matrix. - norm of a matrix. - singular value decomposition of a matrix.
+
  
 
|-
 
|-
−
| 0:31
+
| 00:31
 
| Before beginning this tutorial,we would suggest you to complete the tutorial on "Getting started with lists", "Getting started with arrays", "Accessing parts of arrays".
 
| Before beginning this tutorial,we would suggest you to complete the tutorial on "Getting started with lists", "Getting started with arrays", "Accessing parts of arrays".
  
 
|-
 
|-
−
|  0:42
+
|  00:42
−
| Let us start our ipython interpreter with pylab loaded
+
| Let us start our ipython interpreter by pylab loaded
  
 
|-
 
|-
−
|0:47
+
|00:47
−
|Type ipython hypen pylab  
+
|Type ipython hypen pylab in terminal
  
 
|-
 
|-
−
|  0:52
+
|  00:52
 
| All matrix operations are done using arrays.
 
| All matrix operations are done using arrays.
 
   
 
   
 
|-
 
|-
−
|  0:55
+
|  00:55
−
| Thus all the operations on arrays are valid on matrices also.  
+
| Thus all the operations on arrays are valid on matrices only.  
  
 
|-
 
|-
−
|  1:00
+
|  01:00
 
| A matrix may be created as,
 
| A matrix may be created as,
  
 
|-
 
|-
−
| 1:02
+
| 01:02
 
|Type in the terminal m1 = array within bracket and square bracket 1 comma 2 comma 3 comma 4 hit enter.
 
|Type in the terminal m1 = array within bracket and square bracket 1 comma 2 comma 3 comma 4 hit enter.
  
 
|-
 
|-
−
|  1:16
+
|  01:16
−
| Using the method <tt>shape</tt>, we can find out the shape or size of the matrix,
+
| Using the method shape , we can find out the shape or size of the matrix,
  
 
|-
 
|-
−
|1:20
+
|01:20
 
|Type m1 dot shape and hit enter
 
|Type m1 dot shape and hit enter
  
 
|-
 
|-
−
|1:27
+
|01:27
 
|We can see the output
 
|We can see the output
  
 
|-
 
|-
−
|  1:29
+
|  01:29
 
| Since it is a one row four column matrix it returned a tuple, one by four.
 
| Since it is a one row four column matrix it returned a tuple, one by four.
  
 
|-
 
|-
−
|  1:46
+
|  01:46
 
| A list can also be converted to a matrix as follows,
 
| A list can also be converted to a matrix as follows,
−
 
  
 
|-
 
|-
−
|1:50
+
|01:50
−
|Type l1 = within square bracket square bracket 1 comma 2 comma 3 comma 4 comma in another square bracket 5 comma 6 comma 7 comma 8  
+
|Type l1 = within square bracket square bracket 1 comma 2 comma 3 comma 4 comma in another square bracket 5 comma 6 comma 7 comma 8. Type m2 = array within bracket 11
−
Type m2 = array within bracket 11
+
  
 
|-
 
|-
−
|2:28
+
|02:28
 
|Sorry you have to type l1 array .
 
|Sorry you have to type l1 array .
  
 
|-
 
|-
−
| 2:35
+
| 02:35
 
| Pause the video here, try out the following exercise and resume the video.
 
| Pause the video here, try out the following exercise and resume the video.
  
 
|-
 
|-
−
| 2:43
+
| 02:43
 
| Create a two dimensional matrix m3 of order 2 by 4 with elements 5, 6, 7, 8, 9, 10, 11, 12.
 
| Create a two dimensional matrix m3 of order 2 by 4 with elements 5, 6, 7, 8, 9, 10, 11, 12.
  
 
|-
 
|-
−
| 2:51
+
| 02:51
 
| Switch to terminal for solution.  
 
| Switch to terminal for solution.  
  
 
|-
 
|-
−
|  2:54
+
|  02:54
 
| m3 can be created as,
 
| m3 can be created as,
  
 
|-
 
|-
−
|2:56
+
|02:56
−
|Type m3 = array within closing bracket within square bracket square bracket 5 comma 6 comma 7 comma 8 comma in another square bracket 9 comma 10 comma 11 comma 12
+
|Type m3 = array within closing bracket inside square bracket square bracket 5 comma 6 comma 7 comma 8 comma in another square bracket 9 comma 10 comma 11 comma 12
  
 
|-
 
|-
−
| 3:31
+
| 03:31
 
| Let us now move to matrix operations.
 
| Let us now move to matrix operations.
  
 
|-
 
|-
−
| 3:34  
+
| 03:34  
 
| We can do matrix addition and subtraction easily.
 
| We can do matrix addition and subtraction easily.
  
 
|-
 
|-
−
| 3:37  
+
| 03:37  
 
| m3+m2 does element by element addition, that is matrix addition.  
 
| m3+m2 does element by element addition, that is matrix addition.  
  
 
|-
 
|-
−
| 3:43  
+
| 03:43  
 
| Note that both the matrices should be of the same order.
 
| Note that both the matrices should be of the same order.
  
 
|-
 
|-
−
|3:47
+
|03:47
 
|Type m3+m2 and hit enter so you can see the output.
 
|Type m3+m2 and hit enter so you can see the output.
  
 
|-
 
|-
−
| 3:55
+
| 03:55
 
| Similarly,m3 minus m2 does matrix subtraction, that is element by element subtraction.
 
| Similarly,m3 minus m2 does matrix subtraction, that is element by element subtraction.
  
 
|-
 
|-
−
|4:02
+
|04:02
 
|You can try out by typing m3 minus m2
 
|You can try out by typing m3 minus m2
  
 
|-
 
|-
−
| 4:09
+
| 04:09
 
| Now let us try,matrix multiplication
 
| Now let us try,matrix multiplication
  
 
|-
 
|-
−
|4:13
+
|04:13
 
|Type m3 star m2
 
|Type m3 star m2
  
 
|-
 
|-
−
|  4:20
+
|  04:20
−
| Note that in arrays <tt>m3 star m2</tt> does element wise multiplication and not matrix multiplication,
+
| Note that in arrays m3 star m2 does element wise multiplication and not matrix multiplication,
  
 
|-
 
|-
−
|  4:28
+
|  04:28
−
| Matrix multiplication in matrices are done using the function <tt>dot()</tt>
+
| Matrix multiplication in matrices are done using the function dot()
  
 
|-
 
|-
−
|4:37
+
|04:37
 
|Type dot within bracket m3 comma m2 and hit enter.  
 
|Type dot within bracket m3 comma m2 and hit enter.  
  
 
|-
 
|-
−
|4:47
+
|04:47
 
|So we can see error value show in the command.
 
|So we can see error value show in the command.
  
 
|-
 
|-
−
| 4:50
+
| 04:50
 
| Due to size mismatch, the multiplication could not be done and it returned an error.
 
| Due to size mismatch, the multiplication could not be done and it returned an error.
  
 
|-
 
|-
−
|  4:56
+
|  04:56
 
| Now let us see an example for matrix multiplication.
 
| Now let us see an example for matrix multiplication.
  
 
|-
 
|-
−
|  5:00
+
|  05:00
 
| For doing matrix multiplication we need to have two matrices of the order n by m and m by r and the resulting matrix will be of the order n by r.
 
| For doing matrix multiplication we need to have two matrices of the order n by m and m by r and the resulting matrix will be of the order n by r.
  
 
|-
 
|-
−
|  5:11
+
|  05:11
 
| Thus let us first create two matrices which are compatible for multiplication.
 
| Thus let us first create two matrices which are compatible for multiplication.
  
 
|-
 
|-
−
|5:16
+
|05:16
 
|Type m1.shape and hit enter
 
|Type m1.shape and hit enter
  
 
|-
 
|-
−
| 5:24
+
| 05:24
 
| matrix m1 is of the shape one by four,
 
| matrix m1 is of the shape one by four,
  
 
|-
 
|-
−
|  5:28
+
|  05:28
 
| let us create another one, of the order four by two,
 
| let us create another one, of the order four by two,
  
 
|-
 
|-
−
|5:33
+
|05:33
−
|So type m4 = array in closing bracket within square bracket within square bracket 1 comma 2 comma  within square bracket 3 comma 4 comma  within square bracket 5 comma 6 within square bracket comma  within square bracket 7 comma 8  
+
|So type m4 = array in closing bracket within square bracket within square bracket 1 comma 2 comma  within square bracket 3 comma 4 comma  within square bracket 5 comma 6 within square bracket comma  within square bracket 7 comma 8. Type dot within bracket m1 comma  m4
−
Type dot within bracket m1 comma  m4
+
  
 
|-
 
|-
−
| 6:10
+
| 06:10
−
| Thus the <tt>dot()</tt> function is used for matrix multiplication.
+
| Thus the dot() function is used for matrix multiplication.
  
 
|-
 
|-
−
| 6:15
+
| 06:15
−
| As we already learnt in arrays, the function <tt>identity()</tt> which creates an identity matrix of the order n by n,
+
| As we already learnt in arrays, the function identity() which creates an identity matrix of the order n by n,
  
 
|-
 
|-
−
|  6:24
+
|  06:24
−
| the function <tt>zeros()</tt> which creates a matrix of the order m by n with all zeros,
+
| the function zeros() which creates a matrix of the order m by n with all zeros,
  
 
|-
 
|-
−
|  6:30
+
|  06:30
−
| the function <tt>zeros like function()</tt> which creates a matrix with zeros with the shape of the matrix passed,
+
| the function zeros like function() which creates a matrix with zeros with the shape of the matrix passed,
  
 
|-
 
|-
−
|  6:39
+
|  06:39
−
| the function <tt>ones()</tt> which creates a matrix of order m by n with all ones,
+
| the function ones() which creates a matrix of order m by n with all ones,
  
 
|-
 
|-
−
|  6:47
+
|  06:47
−
| the function <tt>ones underscore like()</tt> which creates a matrix with ones with the shape of the matrix passed;
+
| the function ones underscore like() which creates a matrix with ones with the shape of the matrix passed;
  
 
|-
 
|-
−
|  6:53
+
|  06:53
 
| all these functions can also be used with matrices.
 
| all these functions can also be used with matrices.
  
 
|-
 
|-
−
| 6:57
+
| 06:57
 
|So now Let us see, how to find out the transpose of a matrix we can do,
 
|So now Let us see, how to find out the transpose of a matrix we can do,
  
 
|-
 
|-
−
|7:03
+
|07:03
−
|Type print m4
+
|Type print m4, m4 dot T
−
  m4 dot T
+
  
 
|-
 
|-
−
| 7:14
+
| 07:14
 
| As you saw, Matrix name dot capital T will give the transpose of a matrix
 
| As you saw, Matrix name dot capital T will give the transpose of a matrix
  
 
|-
 
|-
−
| 7:21
+
| 07:21
 
|Pause the video here, try out the following exercise and resume the video.
 
|Pause the video here, try out the following exercise and resume the video.
  
 
|-
 
|-
−
| 7:26
+
| 07:26
 
| Find out the Frobenius norm of inverse of a 4 by 4 matrix, the matrix being,
 
| Find out the Frobenius norm of inverse of a 4 by 4 matrix, the matrix being,
  
 
|-
 
|-
−
|7:33  
+
|07:33  
−
|m5 = arange within bracket 1 comma 17 dot reshape within bracket 4 comma 4
+
|m5 = arrange within bracket 1 comma 17 dot reshape to 4 comma 4
  
 
|-
 
|-
−
| 7:44
+
| 07:44
 
|The Frobenius norm of a matrix is defined as, the square root of the sum of the absolute squares of its elements
 
|The Frobenius norm of a matrix is defined as, the square root of the sum of the absolute squares of its elements
  
 
|-
 
|-
−
| 7:54
+
| 07:54
 
| Switch to terminal for the solution
 
| Switch to terminal for the solution
  
 
|-
 
|-
−
| 7:58  
+
| 07:58  
 
| Let us create the matrix m5 by using the data provided in the question
 
| Let us create the matrix m5 by using the data provided in the question
  
 
|-
 
|-
−
|8:03
+
|08:03
−
|Type m5 = arange within bracket 1 comma 17.reshape within square bracket 4 comma 4
+
|Type m5 = arrange within bracket 1 comma 17.reshape within square bracket 4 comma 4. Then type print m5
−
  print m5
+
  
 
|-
 
|-
−
|  8:20
+
|  08:20
 
| The inverse of a matrix A, A raise to minus one, is also called the reciprocal matrix, such that A multiplied by A inverse will give 1.
 
| The inverse of a matrix A, A raise to minus one, is also called the reciprocal matrix, such that A multiplied by A inverse will give 1.
  
 
|-
 
|-
−
|  8:33
+
|  08:33
 
| The Frobenius norm of a matrix is defined as square root of sum of squares of elements in the matrix.  
 
| The Frobenius norm of a matrix is defined as square root of sum of squares of elements in the matrix.  
  
 
|-
 
|-
−
|  8:41
+
|  08:41
−
| The inverse of a matrix can be found using the function <tt>inv(A)</tt>.
+
| The inverse of a matrix can be found using the function   inv(A) .
  
 
|-
 
|-
−
| 8:47
+
| 08:47
 
| Type in the terminal im5 = inv  within bracket m5  
 
| Type in the terminal im5 = inv  within bracket m5  
  
 
|-
 
|-
−
| 8:57
+
| 08:57
−
| And the Frobenius norm of the matrix <tt>im5</tt> can be found out as,
+
| And the Frobenius norm of the matrix im5 can be found out as, sum = 0
−
sum = 0
+
−
 
+
 
for each in im5 dot flatten function():
 
for each in im5 dot flatten function():
−
    sum plus= each star each
+
sum plus= each star each
 
print sqrt  within bracket sum
 
print sqrt  within bracket sum
  
 
|-
 
|-
−
| 9:52
+
| 09:52
 
| Thus we have successfully obtained the Frobenius norm of the matrix m5
 
| Thus we have successfully obtained the Frobenius norm of the matrix m5
  
 
|-
 
|-
−
|  9:58
+
|  09:58
 
| Pause the video here, try out the following exercise and resume the video.
 
| Pause the video here, try out the following exercise and resume the video.
  
Line 295: Line 288:
 
|-
 
|-
 
|10:20
 
|10:20
−
|sum underscore rows = within square bracket  
+
|sum underscore rows = square bracket  
 
for i in im5 colon
 
for i in im5 colon
−
    sum underscore rows.append  within bracket abs  within bracket i.sum()
+
sum underscore rows.append  within bracket abs  within bracket i.sum()
 
print max within square bracket sum underscore rows
 
print max within square bracket sum underscore rows
  
Line 306: Line 299:
 
|-
 
|-
 
| 11:10
 
| 11:10
−
| The norm of a matrix can be found out using the method <tt>norm()</tt>.
+
| The norm of a matrix can be found out using the method norm().
  
 
|-
 
|-
Line 330: Line 323:
 
|-
 
|-
 
|  11:55
 
|  11:55
−
| Read the documentation of <tt>norm</tt> to read up more about <tt>ord</tt> and the possible type of norms the norm function produces.
+
| Read the documentation of norm to read up more about ord and the possible type of norms the norm function produces.
  
 
|-
 
|-
Line 338: Line 331:
 
|-
 
|-
 
| 12:11  
 
| 12:11  
−
|The determinant of a square matrix can be obtained by using the function <tt>det()</tt> and the determinant of m5 can be found out as,
+
|The determinant of a square matrix can be obtained by using the function det() and the determinant of m5 can be found out as,
  
 
|-
 
|-
Line 354: Line 347:
 
|-
 
|-
 
|12:34
 
|12:34
−
| The eigen values and eigen vector of a square matrix can be computed using the function <tt>eig()</tt> and <tt>eigvals()</tt>.
+
| The eigen values and eigen vector of a square matrix can be computed using the function eig() and eigvals().
  
 
|-
 
|-
Line 374: Line 367:
 
|-
 
|-
 
|  13:11
 
|  13:11
−
| Thus eigen values are given by,eig  within bracket m5  within square bracket 0  
+
| Thus eigen values are given by,eig  within bracket m5  and in square bracket 0  
  
 
|-
 
|-
Line 382: Line 375:
 
|-
 
|-
 
| 13:44  
 
| 13:44  
−
| The eigen values can also be computed using the function <tt>eigvals()</tt> as,
+
| The eigen values can also be computed using the function eigvals() as,
  
 
|-
 
|-
Line 400: Line 393:
 
| Then there exists a factorization of the form
 
| Then there exists a factorization of the form
  
−
M = USigma V star
+
M = U Sigma V star
  
 
|-
 
|-
 
|  14:25
 
|  14:25
−
| where U is an (m by m) unitary matrix over K, the matrix Sigma is an (m by n) diagonal matrix and with non-negative real numbers on the diagonal, and V* is an (n by n) unitary matrix over K,which denotes the conjugate transpose of V.
+
| where U is an (m by m) unitary matrix over K, the matrix Sigma is an (m by n) diagonal matrix and the non-negative real numbers on the diagonal, and V* is an (n by n) unitary matrix over K,which denotes the conjugate transpose of V.
  
 
|-
 
|-
Line 432: Line 425:
 
|-
 
|-
 
| 15:22  
 
| 15:22  
−
| In this tutorial, we have learnt to, 1. Create matrices using arrays.
+
| In this tutorial, we have learnt to, Create matrices using arrays.
  
 
|-
 
|-
 
|  15:25
 
|  15:25
−
| 2. Add,subtract and multiply the elements of matrix.
+
| Add,subtract and multiply the elements of matrix.
  
 
|-
 
|-
 
|  15:28
 
|  15:28
−
| 3. Find out the inverse of a matrix,using the function <tt>inv()</tt>.
+
|Find out the inverse of a matrix,using the function inv() .
  
 
|-
 
|-
 
|  15:32
 
|  15:32
−
| 4. Use the function <tt>det()</tt> to find the determinant of a matrix.
+
| Use the function det() to find the determinant of a matrix.
  
 
|-
 
|-
 
|  15:36
 
|  15:36
−
| 5. Calculate the norm of a matrix using the for loop and also using the function <tt>norm()</tt>.
+
| Calculate the norm of a matrix using the for loop and also using the function norm()  
−
 
+
 
|-
 
|-
 
|  15:43
 
|  15:43
−
| 6. Finding out the eigen vectors and eigen values of a matrix, using functions <tt>eig()</tt> and <tt>eigvals()</tt>.
+
| Finding out the eigen vectors and eigen values of a matrix, using functions eig()   and eigvals()  
  
 
|-
 
|-
 
|  15:50
 
|  15:50
−
| 7. Calculate singular value decomposition(SVD) of a matrix using the function <tt>svd()</tt>.
+
| Calculate singular value decomposition(SVD) of a matrix using the function svd() .
  
 
|-
 
|-
Line 464: Line 456:
 
|-
 
|-
 
|  16:01
 
|  16:01
−
| 1. A and B are two array objects. Element wise multiplication in matrices are done by,
+
| A and B are two array objects. Element wise multiplication in matrices are done by,
−
    A * B
+
A * B
−
  <tt>multiply within bracket A comma  B</tt>
+
multiply within bracket A comma  B
−
  <tt>dot within bracket A comma  B</tt>
+
dot within bracket A comma  B  
−
  <tt>element underscore multiply within bracket A comma B</tt>
+
element underscore multiply within bracket A comma B
  
 
|-
 
|-
 
|  16:19
 
|  16:19
−
| 2. <tt><nowiki>eig within bracket A within square bracket 1</nowiki></tt> and <tt>eigvals within bracket A</tt> are the same.Is it  True or False?
+
| eig within bracket A within square bracket 1 and eigvals within bracket A are the same.Is it  True or False?
  
 
|-
 
|-
 
|  16:31
 
|  16:31
−
| 3. <tt>norm  within bracket A comma ord= within is equal to fro</tt> is the same as <tt>norm within bracket A</tt>. Is it True or False?
+
|norm  within bracket A comma ord= within is equal to fro is the same as norm within bracket A . Is it True or False?
  
 
|-
 
|-
Line 484: Line 476:
 
|-
 
|-
 
|  16:47
 
|  16:47
−
| 1. Element wise multiplication between two matrices, A and B is done as, A into B
+
| Element wise multiplication between two matrices, A and B is done as, A into B
  
 
|-
 
|-
 
| 16:53
 
| 16:53
−
| 2. False. <tt><nowiki>eig within bracket A within square bracket 0</nowiki></tt> and <tt>eigvals  within bracket A</tt> are same, that is both will give the eigen values of matrix A.
+
| False. eig within bracket A within square bracket 0and eigvals  within bracket A are same, that is both will give the eigen values of matrix A.
  
 
|-
 
|-
 
|  17:06
 
|  17:06
−
|3. <tt>norm within bracket A comma ord=is equal to fro</tt> and <tt>norm(A)</tt> are same, since the order=is equal to fro stands for Frobenius norm.
+
|norm within bracket A comma ord=is equal to fro and norm(A) are same, since the order=is equal to fro stands for Frobenius norm.
  
 
|-
 
|-

Latest revision as of 17:11, 20 February 2017

Time Narration
00:01 Hello friends and welcome to the tutorial on 'Matrices'.
00:05 At the end of this tutorial, you will be able to,

Create matrices using data. Create matrices from lists. Do basic matrix operations like addition,multiplication. Perform operations to find out the -- - inverse of a matrix. - determinant of a matrix. - eigen values and eigen vectors of a matrix. - norm of a matrix. - singular value decomposition of a matrix.

00:31 Before beginning this tutorial,we would suggest you to complete the tutorial on "Getting started with lists", "Getting started with arrays", "Accessing parts of arrays".
00:42 Let us start our ipython interpreter by pylab loaded
00:47 Type ipython hypen pylab in terminal
00:52 All matrix operations are done using arrays.
00:55 Thus all the operations on arrays are valid on matrices only.
01:00 A matrix may be created as,
01:02 Type in the terminal m1 = array within bracket and square bracket 1 comma 2 comma 3 comma 4 hit enter.
01:16 Using the method shape , we can find out the shape or size of the matrix,
01:20 Type m1 dot shape and hit enter
01:27 We can see the output
01:29 Since it is a one row four column matrix it returned a tuple, one by four.
01:46 A list can also be converted to a matrix as follows,
01:50 Type l1 = within square bracket square bracket 1 comma 2 comma 3 comma 4 comma in another square bracket 5 comma 6 comma 7 comma 8. Type m2 = array within bracket 11
02:28 Sorry you have to type l1 array .
02:35 Pause the video here, try out the following exercise and resume the video.
02:43 Create a two dimensional matrix m3 of order 2 by 4 with elements 5, 6, 7, 8, 9, 10, 11, 12.
02:51 Switch to terminal for solution.
02:54 m3 can be created as,
02:56 Type m3 = array within closing bracket inside square bracket square bracket 5 comma 6 comma 7 comma 8 comma in another square bracket 9 comma 10 comma 11 comma 12
03:31 Let us now move to matrix operations.
03:34 We can do matrix addition and subtraction easily.
03:37 m3+m2 does element by element addition, that is matrix addition.
03:43 Note that both the matrices should be of the same order.
03:47 Type m3+m2 and hit enter so you can see the output.
03:55 Similarly,m3 minus m2 does matrix subtraction, that is element by element subtraction.
04:02 You can try out by typing m3 minus m2
04:09 Now let us try,matrix multiplication
04:13 Type m3 star m2
04:20 Note that in arrays m3 star m2 does element wise multiplication and not matrix multiplication,
04:28 Matrix multiplication in matrices are done using the function dot()
04:37 Type dot within bracket m3 comma m2 and hit enter.
04:47 So we can see error value show in the command.
04:50 Due to size mismatch, the multiplication could not be done and it returned an error.
04:56 Now let us see an example for matrix multiplication.
05:00 For doing matrix multiplication we need to have two matrices of the order n by m and m by r and the resulting matrix will be of the order n by r.
05:11 Thus let us first create two matrices which are compatible for multiplication.
05:16 Type m1.shape and hit enter
05:24 matrix m1 is of the shape one by four,
05:28 let us create another one, of the order four by two,
05:33 So type m4 = array in closing bracket within square bracket within square bracket 1 comma 2 comma within square bracket 3 comma 4 comma within square bracket 5 comma 6 within square bracket comma within square bracket 7 comma 8. Type dot within bracket m1 comma m4
06:10 Thus the dot() function is used for matrix multiplication.
06:15 As we already learnt in arrays, the function identity() which creates an identity matrix of the order n by n,
06:24 the function zeros() which creates a matrix of the order m by n with all zeros,
06:30 the function zeros like function() which creates a matrix with zeros with the shape of the matrix passed,
06:39 the function ones() which creates a matrix of order m by n with all ones,
06:47 the function ones underscore like() which creates a matrix with ones with the shape of the matrix passed;
06:53 all these functions can also be used with matrices.
06:57 So now Let us see, how to find out the transpose of a matrix we can do,
07:03 Type print m4, m4 dot T
07:14 As you saw, Matrix name dot capital T will give the transpose of a matrix
07:21 Pause the video here, try out the following exercise and resume the video.
07:26 Find out the Frobenius norm of inverse of a 4 by 4 matrix, the matrix being,
07:33 m5 = arrange within bracket 1 comma 17 dot reshape to 4 comma 4
07:44 The Frobenius norm of a matrix is defined as, the square root of the sum of the absolute squares of its elements
07:54 Switch to terminal for the solution
07:58 Let us create the matrix m5 by using the data provided in the question
08:03 Type m5 = arrange within bracket 1 comma 17.reshape within square bracket 4 comma 4. Then type print m5
08:20 The inverse of a matrix A, A raise to minus one, is also called the reciprocal matrix, such that A multiplied by A inverse will give 1.
08:33 The Frobenius norm of a matrix is defined as square root of sum of squares of elements in the matrix.
08:41 The inverse of a matrix can be found using the function inv(A) .
08:47 Type in the terminal im5 = inv within bracket m5
08:57 And the Frobenius norm of the matrix im5 can be found out as, sum = 0

for each in im5 dot flatten function(): sum plus= each star each print sqrt within bracket sum

09:52 Thus we have successfully obtained the Frobenius norm of the matrix m5
09:58 Pause the video here, try out the following exercise and resume the video.
10:04 Find out the infinity norm of the matrix im5.
10:08 The infinity norm of a matrix is defined as the maximum value of sum of the absolute of elements in each row.
10:16 Switch to terminal for the solution
10:20 sum underscore rows = square bracket

for i in im5 colon sum underscore rows.append within bracket abs within bracket i.sum() print max within square bracket sum underscore rows

11:01 Well! to find out the Frobenius norm and Infinity norm we have an even easier method, and let us see that now.
11:10 The norm of a matrix can be found out using the method norm().
11:19 In order to find out the Frobenius norm of the matrix im5, we do,
11:25 In the terminal type norm within bracket im5 and hit enter
11:34 And to find out the Infinity norm of the matrix im5, we do,
11:39 norm within bracket im5,ord=inf
11:51 This is easier when compared to the code we wrote.
11:55 Read the documentation of norm to read up more about ord and the possible type of norms the norm function produces.
12:04 Now let us find out the determinant of a the matrix m5.
12:11 The determinant of a square matrix can be obtained by using the function det() and the determinant of m5 can be found out as,
12:20 So type det within bracket m5
12:26 Hence we get the determinant.
12:29 Let us now move on to eigen vectors and eigen values
12:34 The eigen values and eigen vector of a square matrix can be computed using the function eig() and eigvals().
12:46 Let us find out the eigen values and eigen vectors of the matrix m5. We find them as,
12:53 Typing eig within bracket m5 in the terminal.
13:02 Note that it returned a tuple of two matrices.
13:06 The first element in the tuple are the eigen values and the second element in the tuple are the eigen vectors.
13:11 Thus eigen values are given by,eig within bracket m5 and in square bracket 0
13:30 and the eigen vectors are given by,eig within bracket m5 within square bracket 1
13:44 The eigen values can also be computed using the function eigvals() as,
13:50 Typing on the terminal eigvals within bracket m5
13:58 Now let us learn how to do the singular value decomposition or S V D of a matrix.
14:06 Suppose M is an m (cross) in matrix, whose entries come from the field K, which is either the field of real numbers or the field of complex numbers.
14:18 Then there exists a factorization of the form

M = U Sigma V star

14:25 where U is an (m by m) unitary matrix over K, the matrix Sigma is an (m by n) diagonal matrix and the non-negative real numbers on the diagonal, and V* is an (n by n) unitary matrix over K,which denotes the conjugate transpose of V.
14:53 Such a factorization is called the singular-value decomposition of M.
14:58 The SVD of matrix m5 can be found as
15:01 So now open the terminal and type svd within brackets m5
15:09 Notice that it returned a tuple of 3 elements.
15:12 The first one U the next one Sigma and the third one V star
15:19 This brings us to the end of the end of this tutorial.
15:22 In this tutorial, we have learnt to, Create matrices using arrays.
15:25 Add,subtract and multiply the elements of matrix.
15:28 Find out the inverse of a matrix,using the function inv() .
15:32 Use the function det() to find the determinant of a matrix.
15:36 Calculate the norm of a matrix using the for loop and also using the function norm()
15:43 Finding out the eigen vectors and eigen values of a matrix, using functions eig() and eigvals()
15:50 Calculate singular value decomposition(SVD) of a matrix using the function svd() .
15:58 Here are some self assessment questions for you to solve
16:01 A and B are two array objects. Element wise multiplication in matrices are done by,

A * B multiply within bracket A comma B dot within bracket A comma B element underscore multiply within bracket A comma B

16:19 eig within bracket A within square bracket 1 and eigvals within bracket A are the same.Is it True or False?
16:31 norm within bracket A comma ord= within is equal to fro is the same as norm within bracket A . Is it True or False?
16:43 Look at the answers,
16:47 Element wise multiplication between two matrices, A and B is done as, A into B
16:53 False. eig within bracket A within square bracket 0and eigvals within bracket A are same, that is both will give the eigen values of matrix A.
17:06 norm within bracket A comma ord=is equal to fro and norm(A) are same, since the order=is equal to fro stands for Frobenius norm.
17:22 Hence answer is true.
17:26 Hope you have enjoyed this tutorial and found it useful.
17:30 Thank you!

Contributors and Content Editors

Gaurav, Minal, PoojaMoolya, Sneha

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