In the previous section we used augmented matrices to denote a system of linear equations.  In this section we’re going to start looking at matrices in more generality.  A matrix is nothing more than a rectangular array of numbers and each of the numbers in the matrix is called an entry.  Here are some examples of matrices.

 

 

The size of a matrix with n rows and m columns is denoted by .  In denoting the size of a matrix we always list the number of rows first and the number of columns second.

 

Example 1  Give the size of each of the matrices above.   Solution                                                                          In this matrix the number of rows is equal to the number of columns.  Matrices that have the same number of rows as columns are called square matrices.                                                       This matrix has a single column and is often called a column matrix.                                       This matrix has a single row and is often called a row matrix.                                                        Often when dealing with  matrices we will drop the surrounding brackets and just write -2.

 

Note that sometimes column matrices and row matrices are called column vectors and row vectors respectively.  We do need to be careful with the word vector however as in later chapters the word vector will be used to denote something much more general than a column or row matrix.  Because of this we will, for the most part, be using the terms column matrix and row matrix when needed instead of the column vector and row vector.

 

There are a lot of notational issues that we’re going to have to get used to in this class.  First, upper case letters are generally used to refer to matrices while lower case letters generally are used to refer to numbers.  These are general rules, but as you’ll see shortly there are exceptions to them, although it will usually be easy to identify those exceptions when they happen.

 

We will often need to refer to specific entries in a matrix and so we’ll need a notation to take care of that.  The entry in the i^th row and j^th column of the matrix A is denoted by,

In the first notation the lower case letter we use to denote the entries of a matrix will always match with the upper case letter we use to denote the matrix.  So the entries of the matrix B will be denoted by . 

 

In both of these notations the first (left most) subscript will always give the row the entry is in and the second (right most) subscript will always give the column the entry is in.  So,  will be the entry in the 4^th row and 9^th column of C (which is assumed to be a matrix since it’s an upper case letter…).

 

Using the lower case notation we can denote a general  matrix, A, as follows,

We don’t generally subscript the size of the matrix as we did in the second case, but on occasion it may be useful to make the size clear and in those cases we tend to subscript it as shown in the second case.

 

The notation above for a general matrix is fairly cumbersome so we’ve also got some much more compact notation that we’ll use when we can.  When possible we’ll use the following to denote a general matrix.

The first two we tend to use when we need to talk about the general entry of a matrix (such as certain formulas) but don’t really care what that entry is.  Also, we’ll denote the size if it’s important or needed for whatever we’re doing, but otherwise we’ll not bother with the size.  The third notation is really nothing more than the standard notation with the size denoted.  We’ll use this only when we need to talk about a matrix and the size is important but the entries aren’t.  We won’t run into this one too often, but we will on occasion.

 

We will be dealing extensively with column and row matrices in later chapters/sections so we need to take care of some notation for those.  There are the main exception to the upper case/lower case convention we adopted earlier for matrices and their entries.  Column and row matrices tend to be denoted with a lower case letter that has either been bolded or has an arrow over it as follows,

In written documents, such as this, column and row matrices tend to be in bold face while on the chalkboard of a classroom they tend to get arrows written over them since it’s often difficult on a chalkboard to differentiate a letter that’s in bold from one that isn’t.

 

Also, notice with column and row matrices the entries are still denoted with lower case letters that match the letter that represents the matrix and in this case since there is either a single column or a single row there was no reason to double subscript the entries.

 

Next we need to get a quick definition out of the way for square matrices.  Recall that a square matrix is a matrix whose size is  (i.e. it has the same number of rows as columns).  In a square matrix the entries  (see the shaded portion of the matrix below) are called the main diagonal.

 

The next topic that we need to discuss in this section is that of partitioned matrices and submatrices.  Any matrix can be partitioned into smaller submatrices simply by adding in horizontal and/or vertical lines between selected rows and/or columns. 

 

Example 2  Here are several partitions of a general  matrix. (a)   In this case we partitioned the matrix into four submatrices.  Also notice that we simplified the matrix into a more compact form and in this compact form we’ve mixed and matched some of our notation.  The partitioned matrix can be thought of as a smaller matrix with four entries, except this time each of the entries are matrices instead of numbers and so we used capital letters to represent the entries and subscripted each one with the location in the partitioned matrix.    Be careful not to confuse the location subscripts on each of the submatrices with the size of each submatrix.  In this case  is a  sub matrix of A,  is a  sub matrix of A,  is a  sub matrix of A, and  is a  sub matrix of A.   (b)   In this case we partitioned A into three column matrices each representing one column in the original matrix.  Again, note that we used the standard column matrix notation (the bold face letters) and subscripted each one with the location in the partitioned matrix.  The  in the partitioned matrix are sometimes called the column matrices of A.   (c)   Just as we can partition a matrix into each of its columns as we did in the previous part we can also partition a matrix into each of its rows. The  in the partitioned matrix are sometimes called the row matrices of A.

 

The previous example showed three of the many possible ways to partition up the matrix.  There are, of course, many other ways to partition this matrix.  We won’t be partitioning up too many matrices here, but we will be doing it on occasion, so it’s a useful idea to remember.  Also note that when we do partition up a matrix into its column/row matrices we will generally put in the bars separating the columns/rows as we’ve done here to indicate that we’ve got a partitioned matrix.

 

To close out this section we’re going to introduce a couple of special matrices that we’ll see show up on occasion. 

 

The first matrix is the zero matrix.  The zero matrix is pretty much what the name implies.  It is an  matrix whose entries are all zeroes.  The notation we’ll use for the zero matrix is  for a general zero matrix or  for a zero column or row matrix.  Here are a couple of zero matrices just so we can say we have some in the notes.

If the size of a column or row zero matrix is important we will sometimes subscript the size on those as well just to make it clear what the size is.  Also, if the size of a full zero matrix is not important or implied from the problem we will drop the size from  and just denote it by 0.

 

The second special matrix we’ll look at in this section is the identity matrix.  The identity matrix is a square  matrix usually denoted by  or just I if the size is unimportant or clear from the context of the problem.  The entries on the main diagonal of the identity matrix are all ones and all the other entries in the identity matrix are zeroes.  Here are a couple of identity matrices.

 

As we’ll see identity matrices will arise fairly regularly.  Here is a nice theorem about the reduced row-echelon form of a square matrix and how it relates to the identity matrix.

 

Theorem 1 If A is an  matrix then the reduced row-echelon form of the matrix will either contain at least one row of all zeroes or it will be , the  identity matrix.

 

Proof : This is a simple enough theorem to prove that we may as well.  Let’s suppose that B is the reduced row-echelon form of the matrix.  If B has at least one row of all zeroes we are done so let’s suppose that B does not have a row of all zeroes.  This means that every row has a leading 1 in it.

 

Now, we know that the leading 1 of a row must be to the right of the leading 1 of the row immediately above it.  Because we are assuming that B is square and doesn’t have any rows of all zeroes we can actually locate each of the leading 1’s in B.

 

First, let’s suppose that the leading 1 in the first row is NOT  (i.e.  ).  The next possible location of the leading 1 in the first row would then be .  So, let’s suppose that this is where the leading 1 is.  So, upon assuming this we can say that B must have the following form.

 

Now, let’s assume the best possible scenario happens.  That is the leading 1 of each of the lower rows is exactly one column to the right of the leading 1 above it.  This however, leads us to instant problems.  Because our first leading 1 is in the second column by the time we reach the n-1^st row our leading 1 will be in the n^th column and this will in turn force the n^th row to be a row of all zeroes which contradicts our initial assumption.  If you’re not sure you believe this consider the  case.

Sure enough a row of all zeroes in the 4^th row.

 

Now, we assumed the best possible scenario for the leading 1’s in the lower rows and ran into problems.  If the leading 1 jumps to the right say 2 columns (or 3 or 4, etc.) we will run into the same kind of problem only we’ll end up with more than one row of all zeroes.

 

Likewise if the leading 1 in the first row is in any of  we will have the same problem.  So, in order to meet the assumption that we don’t have any rows of all zeroes we know that the leading 1 in the first row must be at .

 

Using a similar argument to that above we can see that if the leading 1 on any of the lower rows jumps to the right more than one column we will have a leading 1 in the n^th column prior to hitting the n^th row.  This will in turn force at least the n^th row to be a row of all zeroes which will again contradict our initial assumption.

 

Therefore we know that the leading one in the first row is at  and the only hope of not having a row of all zeroes at the bottom is to have the leading 1’s of a row be exactly one column to the right of the leading 1 of the row above it.  This means that the leading 1 in the second row must be at , the leading 1 in the third row must be at , etc.  Eventually we’ll hit the n^th row and in this row the leading 1 must be at . 

 

Therefore the leading 1’s of B must be on the diagonal and because B is the reduced row-echelon form of A we also know that all the entries above and below the leading 1’s must be zeroes.  This however, is exactly .  Therefore, if B does not have a row of all zeroes in it then we must have that .