Abstract Nonsense

Crushing one theorem at a time

Algebra of Square Matrices (Pt. II)


Point of post: This post is a continuation of this post. We will finish the proof started in the last post and then wrap up our discussion of square matrices as independent algebraic objects.

(continuing the proof at the end of the last post) We now prove that the distributivity property holds. To do this let M,N, and P be as above and note that, by definition, the general entry of M(N+P) is

\displaystyle \sum_{r=1}^{n}\alpha_{i,r}\left(\beta_{r,j}+\gamma_{r,j}\right)=\sum_{r=1}^{n}\alpha_{i,r}\beta_{r,j}+\sum_{r=1}^{n}\alpha_{i,r}\gamma_{r,j}

which is the general term for MN+MP. The other distributivity axiom follows similarly.

 

To prove the scalar distributivity property we merely note that if M and N are as before and \alpha\in F then, by definition the general entry of \alpha(MN) is

\displaystyle \alpha \sum_{r=1}^{n}\alpha_{i,r}\beta_{r,j}=\sum_{r=1}^{n}(\alpha \alpha_{i,r})\beta_{r,j}=\sum_{r=1}^{n}\alpha_{i,r}(\alpha \beta_{r,j})

But, these last two forms are the general entry of (\alpha M)N and M(\alpha N) respectively.

 

It remains to show that I_n is, in fact, a multiplicative identity. To do this it suffices to note that if M is as before then the general entry of MI_n is

\displaystyle \sum_{r=1}^{n}\alpha_{i,r}\delta_{r,j}=\alpha_{i,j}\delta_{j,j}=\alpha_{i,j}

which is the general entry for M. Similarly, the general entry for I_nM is

\displaystyle \sum_{r=1}^{n}\delta_{i,r}\alpha_{r,j}=\delta_{i,i}\alpha_{i,j}=\alpha_{i,j}

which is the general entry for M. It follows that MI_n=I_nM=M as required.

 

Thus, since all the axioms have been verified the conclusion follows. \blacksquare

 

References:

1. Golan, Jonathan S. The Linear Algebra a Beginning Graduate Student Ought to Know. Dordrecht: Springer, 2007. Print.

2. Halmos, Paul R.  Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print

Advertisements

December 13, 2010 - Posted by | Algebra, Halmos, Linear Algebra | ,

No comments yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: