## Algebra of Square Matrices (Pt. I)

**Point of post: **In this post we discuss the concept of the *square matrix algebra of size over a field * in the quest to approach the concept of the relation between matrices and linear transformations in the opposite way in which Halmos views them in sections 36 and 37 of his book. Namely, we start out by defining matrices as being algebraic objects in their own right, and then connect the endomorphism algebra of a vector space canonically with a particular matrix algebra in a canonical way.

*Motivation*

In this post we show how given a field there is a natural way to produce an associative unital algebra of dimension over . This algebra will be the algebra of square matrices (to be defined below) of size . Not only will this algebra serve as another interesting addition to our menagerie of vector spaces (algebras) but will give us a fascinating and useful means to look at the endomorphism algebra of a vector space is a whole new way.

*Definition of Matrices*

Let be a field and . For we call the square array

a *square matrix of size over *. We denote the set of all such square matrices by . For convenience sake we denote square matrices as either an upper case letter, such as , or we use the notation (or both). We call the vector the *row *of and the vector the *column *of . We call the *entry *of and we say two matrices are *equal *if they have the same entries. In essence, a matrix can really be thought of as an -tuple of elements of .

*Vector Space Structure*

We now endow with a vector space structure over , namely if and and we define the *sum of and times the scalars and respectively*, denoted by, to be the matrix where . Put more visually

We then define the zero element to be the *zero matrix*, denoted , to be the matrix

and as a special case as the sum of matrices times scalars, for a matrix we define to be , in other words scalars are distributed entry-wise. We leave it to the reader to verify that this structure does, in fact, define a vector space structure on .

We now prove what was hinted at in the definition of matrices, namely that is just a guise for the common space . Put more directly:

**Theorem: ***Let be a field, then:*

**Proof: **

To do this we merely show that is a basis for where where

But this follows directly from the observation that if then

This follows directly from our earlier characterization of isomorphisms since evidently is linear and letting be the usual basis for we see then that which is evidently a bijection between the two bases.

*Algebra Structure of Square Matrices*

We now discuss the much more interesting structure on , namely the multiplicative structure; this structure is much less intuitive. So, if We define the *product of and *, denoted by the concatenation where

Once again, more visually

It is perhaps less obvious with this above multiplication and the definition of the sum of two matrices given in the previous section that is turned into an associative unital algebra, so we’ll prove this

**Theorem: ***Let be a field and let have the vector space structure given in the first section and the multiplicative structure given in the second section. Then, is an associative unital algebra with identity element where (where is, as usual, the Kronecker Delta Symbol)*

**Proof: **We first prove that the multiplication described above is, in fact, associative. To do this we let and note that the general term of is

and the general term of is

Thus, the general term of is

which is the general term for . from where associativity follows. (Proof continued on next post)

**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

[…] 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 […]

Pingback by Algebra of Square Matrices (Pt. II) « Abstract Nonsense | December 13, 2010 |

[…] our last post we saw how to endow ( in disguise) with the structure of an associative unital algebra. We hinted […]

Pingback by Algebra Isomorphism Between Algebra of Square Matrices and the Endomorphism Algebra (Pt. I) « Abstract Nonsense | December 16, 2010 |

[…] read my side-along postings you will probably need to see the series of posts for which this and this are the first posts for […]

Pingback by Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. I) « Abstract Nonsense | December 19, 2010 |

[…] course we know that the space of of matrices over , , is an associative unital algebra with the usual […]

Pingback by The Hilbert-Schmidt Inner Product on Complex Matrix Algebras « Abstract Nonsense | April 5, 2011 |

[…] by where and where . The fact that with these operations is really a ring is the same as the case when is a […]

Pingback by Matrix Rings (Pt. I) « Abstract Nonsense | July 12, 2011 |