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

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 A_2,2 is multiplied by identity matrix I_2,2 then the result will be A_2,2. So, A_2,2 ❌ I_2,2 = A_2,2

A2,2 =

[

I2,2 =

[

A2,2 =

[

A2,2 =

[

2

3

❌

1

0

=

2❌1 ➕ 3❌0

2❌0 ➕ 3❌1

=

2

3

5

4

0

1

5❌1 ➕ 4❌0

5❌0 ➕ 4❌1

5

4

]

]

]

]

 

Identity matrix looks like above I_2,2. 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 I_2,2, I_4,4, I_6,6, I_3,3 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 A_2,2.

A2,2 =	[
		2	3
		4	5
	]

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 =	[
		2	3
		4	5
	]

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 =

[

2

3

.

5/-2

3/2

=

2❌5/-2 ➕ 3❌2

2❌3/2 ➕ 3❌-1

=

1

0

4

5

2

-1

4❌5/-2 ➕ 5❌2

4❌3/2 ➕ 5❌-1

0

1

]

]

]

]

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