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

5 comments :

l'orchidée said...

I just found this post using google and I just wanted to thank you for it. Your blog is awesome and I hope you continue to do this (you straightfowardly explained something in my math homework that I just could not understand, even with the help of my math buddies). Thanks again, hope you continue to do this.

Larry Freeman said...

Thanks very much for the feedback. I am very glad that my blog helped. :-)

Cheers,

-Larry

Anonymous said...

your blog rocks thanks, you da man

Anonymous said...

this is fantastic
helps me with proofs alot!

mc greene said...

I am a PhD econ student who has returned to school after many years of working. This is a lifesaver for picking up the details that many advanced proofs skip. Thanks a million!!!