LINEAR ALGEBRA (Part-2) - Multiplying Matrices

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.

A3,2 =	[
		1	2
		2	3
		3	1
	]

Let's add a scalar value 5 with it.

				A3,2 =	[						A3,2 =	[						A3,2 =	[
						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.

				A3,2 =	[						A3,2 =	[						A3,2 =	[
						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.

				A3,2 =	[						A3,2 =	[						A3,2 =	[
						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.

				A3,2 =	[						A3,2 =	[						A3,2 =	[
						1	2						5➗1	5➗2						5	2.5
5		➗				2	3		=				5➗2	5➗3		=				2.5	1.67
						3	1						5➗3	5➗1						1.67	5
					]							]							]

Standard Multiplication

It is also called Standard Product. The standard multiplication of matrix A and B will be another matrix C. Matrix are represented by row ✖ column. If matrix A is 3✖2 and B is 3✖2 then resultant matrix C will also be 3✖2 matrix. We can represent it by C_3,2 = A_3,2 ⊙ B_3,2

For standard multiplication both matrix have to be of same size. It will be a element wise multiplication. Lets see below

A3,2 =	[						B3,2 =	[						C3,2 =	[						A3,2 =	[
		4	5						1	2						4❌1	5❌2						4	10
		5	6		⊙				2	3		=				5❌2	6❌3		=				10	18
		6	4						3	1						6❌3	4❌1						18	4
	]							]							]							]

Similarly we can also do addition, subtration and division also.

Matrix Multiplication

It is also called Matrix Product. The multiplication of matrix A and B will be another matrix C. Matrix are represented by row ✖ column. If matrix A is 3✖2 and B is 2✖4 then resultant matrix C will be 3✖4 matrix. We can represent it by C_3,4 = AB

For Matrix Multiplication the column size of one matrix have to be equal to row size of another matrix and vice versa. Let's see the multiplication below.

A3,2 =	[						B2,3 =	[							C3,3 =	[									A3,3 =	[
		4	5						1	2	3						4❌1 ➕ 5❌2		4❌2 ➕ 5❌3		4❌3 ➕ 5❌1						14	23	17
		5	6		•				2	3	1		=				5❌1 ➕ 6❌2		5❌2 ➕ 6❌3		5❌3 ➕ 6❌3		=				17	28	33
		6	4					]									6❌1 ➕ 4❌2		6❌2 ➕ 4❌3		6❌1 ➕ 4❌1						14	24	10
	]															]										]

SOURCE

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