## Sunday, February 25, 2007

### Identity Matrix

In a previous blog, I reviewed the basic properties of 2 x 2 matrices. In today's blog, I generalize the definition of the identity matrix and show some of its properties.

Let me start by defining the Identity Matrix In.

Definition 1: In: Identity Matrix for n x n matrix

The Identity Matrix for n is an n x n matrix consists of values [ai,j] where i = the row and j = column where:

i = j → ai,j = 1

i ≠ j → ai,j=0

Here are a few examples:

If An,n is an n x n matrix, we have:

Property 1: An,nIn = An,n

Proof:

(1) Let:

(2) Then, An,nIn =

QED

Property 2: InAn,n = An,n

Proof:

(1) Let:

(2) Then, InAn,n =

QED

If An,m is an n x m matrix, we have:

Property 3: An,mIm=An,m

(1) Let:

(2) Then, An,mIm =

QED

References