LINEAR ALGEBRA (Part-4) - Identity and Inverse Matrices

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LINEAR ALGEBRA (Part-4)

Identity and Inverse Matrices

Identity Matrix

It is a square matrix that do not change any matrix or vector when multiplied with. Means If a matrix A2,2 is multiplied by identity matrix I2,2 then the result will be A2,2. So, A2,2 ❌ I2,2 = A2,2
A2,2 =[     I2,2 =[     A2,2 =[      A2,2 =[  
  23 ❌   10 =   2❌1 ➕ 3❌0 2❌0 ➕ 3❌1 =   23
  54     01     5❌1 ➕ 4❌0 5❌0 ➕ 4❌1     54
 ]      ]      ]       ]  

 

Identity matrix looks like above I2,2. Where diagonally all values are 1 and rest are 0. Identity matrix is a square matrix so its row and column are always same. Like I2,2, I4,4, I6,6, I3,3 etc.

Matrix Inverse

Matrix inverse is a bit complex. Don't worry we will not use this a lot. But we need it to verify some property. We will not go deep into inverse matrix. You can learn that in source (3) given at the end.
Lets consider a matrix A2,2.
A2,2 =[  
  23
  45
 ]  
So the inverse of this matrix will be A-1. We will get the inverse by swapping 2 and 5, put negatives in front of 3 and 4, and divide everything by the determinant (2❌5 - 3❌4). So the inverse will be.
A-1 =    [      A-1 =[   
 1/-2 ❌  5 -3 =   5/-2 -3/-2
      -4 2     -4/-2 2/-2
     ]       ]   

Property

Let's learn another property of matrix. If we multiply a matrix with the inverse of the same matrix we will get an Identity matrix. Means A-1A = I
Lets see the multiplication.
A2,2 =[  
  23
  45
 ]  
So the inverse of this matrix will be A-1.
A-1 =[   
  5/-2 3/2
  2 -1
 ]   
As per the theory we should get A-1A = I. Let's see:
A = [     A-1 =[      I =[      I =[  
  23 .   5/-2 3/2 =   2❌5/-2 ➕ 3❌2 2❌3/2 ➕ 3❌-1 =   10
  45     2 -1     4❌5/-2 ➕ 5❌2 4❌3/2 ➕ 5❌-1     01
 ]      ]       ]       ]  

Property

We will finish this part with another property. Remember the Linear Algebra part-3, property 8 says "The multiplication of a matrix and a vector results in a vector." means Ab=c where A is matrix, b and c is vector.
If we consider this a simple math problem and move A from one side to another we will get b = c/A.
This is actually true for matrix also. means b = A-1c. Lets see How
Ab = c
Multiply both side with A-1
A-1Ab = A-1c
we know from above A-1A = I, so
Ib = A-1c
We also know that multiplying any matrix or vector with identity matrix results in the same matrix or vector. So Ib = b. as a result we get
b = A-1c
The above process depends on calculating A-1 which is not always possible to calculate.
One last important thing A−1 is primarily useful as a theoretical tool, and should not actually be used in practice for most software applications



SOURCE

1) Deep Learning (Ian Goodfellow, Yoshua Bengio, Aaron Courville)

2) https://www.mathsisfun.com/algebra/matrix-inverse.html

3) https://www.mathsisfun.com/algebra/matrix-inverse-minors-cofactors-adjugate.html

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LINEAR ALGEBRA (Part-2) - Multiplying Matrices

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LINEAR ALGEBRA (Part-2) Multiplying Matrices Standard Calculation We can add, subtract, divide or multiply a matrix with a scalar value. These operations will be element wise. Let's consider a 3✖2 Matrix A 3,2 . See below. A 3,2 = [                                     1 2                                     2 3                                     3 1                                     ]                                     Let's add a scalar value 5 with it. A 3,2 = [ A 3,2 = [  A 3,2 = [ 1 2 5✚1  5✚2  6 7 5 ✚ 2 3 =  5✚2 5✚3  = 7 8 3 1 5✚3  5✚1  8 6 ] ]  ] Let's subtract a scalar value 5 with it. A 3,2 = [ A 3,2 = [  A 3,2 = [ 1 2 5➖1  5➖2  4 3 5 ➖ 2 3 =  5➖2 5➖3  = 3 2 3 1 5➖3  5➖1  2 4 ] ]  ] Let's multiply a scalar value 5 with it. A 3,2 = [ A 3,2 = [  A 3,2 = [ 1 2 5❌1  5❌2  5 10 5 ❌ 2 3 = 5❌2 5❌3  = 10 15 3 1 5❌3  5❌1  15 5 ] ]  ] Let's divide a scalar value 5 with it. A 3,2 = [ A 3,2
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What Is derivative? (Part - 1)

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What Is derivative? (Part - 1) Derivative is a simple thing. It’s just the difference of a variable. But how? Think of time, let a person started a journey towards bus stand at time t1 = 3 and reached at t2 = 7. In this case the derivative dt = t2-t1 = 7-3 = 4. Easy! But in real life things are not that easy. Because we have do derive the difference of a variable with respect to another variable unlike our time example where there was only one variable. For example we can see the equation y = x. Here we have to find the difference in y with respect to the change in variable x. In other word we have to find the derivative of y with respect to x which is dy/dx. Fortunately for us this is very easy. Pretty easy right. But we are going to prove it so that we can use the same theory to prove more complex derivatives. So we are going to plot this function into graph. Let’s find some value of x and values of y for the respective value of x below.
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LINEAR ALGEBRA (Part-1) - Scalars, Vectors, Matrices and Tensors

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LINEAR ALGEBRA (Part-1) Scalars, Vectors, Matrices and Tensors Scalars Scalars are a single number. like 5 or 8 or 9 etc. you can say age=23 here 23 is a scalar value. Vectors Vectors are a series/array of elements. They are represented between a square bracket with spaces. Like below. ages = [    23    25    33    54 ] Here, [    23    25    33    54 ] is a Column Vector of ages. The Row Vector of ages will look like ages = [23 25 33 54] each element of a vector can be represented by a subscript number like ages 1 =23 or ages 3 =33 etc. the numbers of a vector do not have to be of same thing like. box = [    2    4    8    "wood" ] here the box vector represents, box 1 = height = 2 inch box 2 = width = 4 inch box 3 = weight = 8 gram box 4 = made of = wood Matrix Matrix is a 2D array of elements. Like below. box = [    2    3    2    4    4    4    5    2    8    12    5
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