LINEAR ALGEBRA (Part-3) - Useful Properties

LINEAR ALGEBRA (Part-3)

Useful Properties

Information

Below, we will see some properties of Matrix. Remember there are many more then listed.

Property 1

A(B + C) = AB + AC

Property 2

A(BC) = (AB)C

Property 3

A(BC) = (AB)C

Property 4

AB ≠ BA
For some cases AB = BA is true. But not for all.

Property 5

A^TB = B^TA

Property 6

(AB)^T = B^TA^T

Property 7

A^TB = (A^TB)^T = B^TA

Property 8

A ❌ b = c

The multiplication of a matrix and a vector results in a vector.
Let's consider 3X2 matrix A and 2X1 vector b. So A ❌ b will result in 3X1 vector c. see below

A3,2 =	[						b2,1 =	[					c3,1 =	[					c3,1 =	[
		1	3						4						1❌4 ➕ 3❌5						19
		2	1		.				5		=				2❌4 ➕ 1❌5		=				13
		3	2					]							3❌4 ➕ 2❌5						22
	]													]						]

Remember the reverse will also hold. means consider 1X2 vector b and 2X3 matrix A. So b ❌ A will result in 1X3 vector c. see below

b1,2 =	[							A2,3 =	[									c1,3 =	[									c1,3 =	[
		4		5		.				1		2		3		=				4❌1 ➕ 5❌3		4❌2 ➕ 5❌1		4❌3 ➕ 5❌2		=				19		13		22
	]									3		1		2					]										]
									]

SOURCE

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

2) https://www.varsitytutors.com/hotmath/hotmath_help/topics/multiplying-vector-by-a-matrix