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The Identity Matrix

Module by: Kenny M. Felder. E-mail the author

Summary: This module introduces the identity matrix and its properties.

When multiplying numbers, the number 1 has a special property: when you multiply 1 by any number, you get that same number back. We can express this property as an algebraic generalization:

1 x = x 1x=x
(1)

The matrix that has this property is referred to as the identity matrix.

Definition of Identity Matrix

The identity matrix, designated as [ I ] [I], is defined by the property: [ A ] [ I ] = [ I ] [ A ] = [ A ] [A][I]=[I][A]=[A]

Note that the definition of [I] stipulates that the multiplication must commute—that is, it must yield the same answer no matter which order you multiply in. This is important because, for most matrices, multiplication does not commute.

What matrix has this property? Your first guess might be a matrix full of 1s, but that doesn’t work:

Table 1
12341234 size 12{ left [ matrix { 1 {} # 2 {} ## 3 {} # 4{} } right ]} {}11111111 size 12{ left [ matrix { 1 {} # 1 {} ## 1 {} # 1{} } right ]} {} = 33773377 size 12{ left [ matrix { 3 {} # 3 {} ## 7 {} # 7{} } right ]} {} so 1 1 1 1 1 1 1 1 size 12{ left [ matrix { 1 {} # 1 {} ## 1 {} # 1{} } right ]} {} is not an identity matrix

The matrix that does work is a diagonal stretch of 1s, with all other elements being 0.

Table 2
12341234 size 12{ left [ matrix { 1 {} # 2 {} ## 3 {} # 4{} } right ]} {}10011001 size 12{ left [ matrix { 1 {} # 0 {} ## 0 {} # 1{} } right ]} {}= 12341234 size 12{ left [ matrix { 1 {} # 2 {} ## 3 {} # 4{} } right ]} {} so 1 0 0 1 1 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} ## 0 {} # 1{} } right ]} {} is the identity for 2x2 matrices
259π2831/28.3259π2831/28.3 size 12{ left [ matrix { 2 {} # 5 {} # 9 {} ## π {} # - 2 {} # 8 {} ## - 3 {} # 1/2 {} # 8 "." 3{} } right ]} {}100010001100010001 size 12{ left [ matrix { 1 {} # 0 {} # 0 {} ## 0 {} # 1 {} # 0 {} ## 0 {} # 0 {} # 1{} } right ]} {} = 259π2831/28.3259π2831/28.3 size 12{ left [ matrix { 2 {} # 5 {} # 9 {} ## π {} # - 2 {} # 8 {} ## - 3 {} # 1/2 {} # 8 "." 3{} } right ]} {} 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} # 0 {} ## 0 {} # 1 {} # 0 {} ## 0 {} # 0 {} # 1{} } right ]} {} is the identity for 3x3 matrices

You should confirm those multiplications for yourself, and also confirm that they work in reverse order (as the definition requires).

Hence, we are led from the definition to:

The Identity Matrix

For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upper-left-hand corner to the lower-right, with all other elements being 0. Non-square matrices do not have an identity. That is, for a non-square matrix [ A ] [A], there is no matrix such that [ A ] [ I ] = [ I ] [ A ] = [ A ] [A][I]=[I][A]=[A].

Why no identity for a non-square matrix? Because of the requirement of commutativity. For a non-square matrix [ A ] [A] you might be able to find a matrix [ I ] [I] such that [ A ] [ I ] = [ A ] [A][I]=[A]; however, if you reverse the order, you will be left with an illegal multiplication.

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