This section is devoted to a couple of special matrices that we could have talked about pretty much anywhere, but due to the desire to keep most of these sections as small as possible they just didn’t fit in anywhere.  However, we’ll need a couple of these in the next section and so we now need to get them out of the way.

 

Diagonal Matrix

This first one that we’re going to take a look at is a diagonal matrix.  A square matrix is called diagonal if it has the following form.

 

In other words, in a diagonal matrix is any matrix in which the only potentially non-zero entries are one the main diagonal.  Any entry off the main diagonal must be zero and note that it is possible to have one or more of the main diagonal entries be zero.

 

We’ve also been dealing with a diagonal matrix already to this point if you think about it a little.  The identity matrix is a diagonal matrix.

 

Here is a nice theorem about diagonal matrices.

 

Theorem 1 Suppose D is a diagonal matrix and  are the entries on the main diagonal.  If one or more of the  ’s are zero then the matrix is singular.  On the other hand if  for all i then the matrix is invertible and the inverse is,

 

Proof : First, recall Theorem 3 from the previous section.  This theorem tells us that if D is row equivalent to the identity matrix then D is invertible and if D is not row equivalent to the identity then D is singular.

 

If none of the  ’s are zero then we can reduce D to the identity simply by dividing each of the rows its diagonal entry (which we can do since we’ve assumed none of them are zero) and so in this case D will be row equivalent to the identity.  Therefore, in this case D is invertible.  We’ll leave it to you to verify that the inverse is what we claim it to be.  You can either compute this directly using the method from the previous section or you can verify that .

 

Now, suppose that at least one of the  is equal to zero.  In this case we will have a row of all zeroes, and because D is a diagonal matrix all the entries above the main diagonal entry in this row will also be zero and so there is no way for us to use elementary row operations to put a 1 into the main diagonal and so in this case D will not be row equivalent to the identity and hence must be singular.

 

Powers of diagonal matrices are also easy to compute.  If D is a diagonal matrix and k is any integer then

 

Triangular Matrix

The next kind of matrix we want to take a look at will be triangular matrices.  In fact there are actually two kinds of triangular matrices.  For an upper triangular matrix the matrix must be square and all the entries below the main diagonal are zero and the main diagonal entries and the entries above it may or may not be zero.  A lower triangular matrix is just the opposite.  The matrix is still a square matrix and all the entries of a lower triangular matrix above the main diagonal are zero and the main diagonal entries and those below it may or may not be zero.

 

Here are the general forms of an upper and lower triangular matrix.

In these forms the  and  may or may not be zero.

 

If we do not care if the matrix is upper or lower triangular we will generally just call it triangular.

 

Note as well that a diagonal matrix can be thought of as both an upper triangular matrix and a lower triangular matrix.

 

Here’s a nice theorem about the invertibility of a triangular matrix.

 

Theorem 2 If A is a triangular matrix with main diagonal entries  then if one or more of the  ’s are zero the matrix will be singular.  On the other hand if  for all i then the matrix is invertible.

 

Here is the outline of the proof.

 

Proof Outline : First assume that  for all i.  In this case we can divide each row by  (since it’s not zero) and that will put a 1 in the main diagonal entry for each row.  Now use the third row operation to eliminate all the non-zero entries above the main diagonal entry for an upper triangular matrix or below it for a lower triangular matrix.  When done with these operations we will have reduced A to the identity matrix.  Therefore, in this case A is row equivalent to the identity and so must be invertible.

 

Now assume that at least one of the  are zero.  In this case we can’t get a 1 in the main diagonal entry just be dividing by  as we did in the first place.  Now, for a second let’s suppose we have an upper triangular matrix.  In this case we could use the third row operation using one of the rows above this to get a 1 into the main diagonal entry, however, this will also put non-zero entries into the entries to the left of this as well.  In other words, we’re not going to be able to reduce A to the identity matrix.  The same type of problem will arise if we’ve got a lower triangular matrix.

 

In this case, A will not be row equivalent to the identity and so will be singular.

 

Here is another set of theorems about triangular matrices that we aren’t going to prove.

 

Theorem 3    (a) The product of lower triangular matrices will be a lower triangular matrix. (b) The product of upper triangular matrices will be an upper triangular matrix. (c) The inverse of an invertible lower triangular matrix will be a lower triangular matrix. (d) The inverse of an invertible upper triangular matrix will be an upper triangular matrix.

 

The proof of these will pretty much follow from how products and inverses are found and so will be left to you to verify.

 

The final kind of matrix that we want to look at in this section is that of a symmetric matrix.  In fact we’ve already seen these in a previous section we just didn’t have the space to investigate them in more detail in that section so we’re going to do it here.

 

For completeness sake we’ll give the definition here again.  Suppose that A is an  matrix, then A will be called symmetric if .

 

Note that the first requirement for a matrix to be symmetric is that the matrix must be square.  Since the size of  will be  there is no way A and  can be equal if A is not square since they won’t have the same size.

 

Example 1  The following matrices are all symmetric.

 

We’ll leave it to you to compute the transposes of each of these and verity that they are in fact symmetric.  Notice with the second matrix (B) above that you can always quickly identify a symmetric matrix by looking at the diagonals off the main diagonal.  The diagonals right above and below the main diagonal consists of the entries -10, 1, 8 are identical.  Likewise, the diagonals two above and below the main diagonal consists of the entries 3, -4 and again are identical.  Finally, the “diagonals” that are three above and below the main diagonal is identical as well.

 

This idea we see in the second matrix above will be true in any symmetric matrix.

 

Here is a nice set of facts about arithmetic with symmetric matrices.

 

Theorem 4 If A and B are symmetric matrices of the same size and c is any scalar then, (a)  is symmetric. (b) cA is symmetric. (c)  is symmetric.

 

Note that the product of two symmetric matrices is probably not symmetric.  To see why this is consider the following.  Suppose both A and B are symmetric matrices of the same size then,

Notice that we used one of the properties of transposes we found earlier in the first step and the fact that A and B are symmetric in the last step.

 

So what this tells us is that unless A and B commute we won’t have  and the product won’t be symmetric.  If A and B do commute then the product will be symmetric.

 

Now, if A is any  matrix then because  will have size  both  and  will be defined and in fact will be square matrices where  has size  and  has size .

 

Here are a couple of quick facts about symmetric matrices.

 

Theorem 5   (a) For any matrix A both  and  are symmetric. (b) If A is an invertible symmetric matrix then  is symmetric. (c) If A is invertible then  and  are both invertible.

 

Proof :

(a) We’ll show that  is symmetric and leave the other to you to verify.  To show that  is symmetric we’ll need to show that .  This is actually quite simple if we recall the various properties of transpose matrices that we’ve got.

(b) In this case all we need is a theorem from a previous section to show that .  Here is the work,

(c) If A is invertible then we also know that  is invertible and since the product of invertible matrices is invertible both  and  are invertible.

 

Let’s finish this section with an example or two illustrating the results of some of the theorems above.

 

Example 2  Given the following matrices compute the indicated quantities.               (a) AB   [Solution] (b)    [Solution] (c)    [Solution] (d)    [Solution]   Solution (a) AB There really isn’t much to do here other than the multiplication and we’ll leave it to you to verify the actual multiplication.                                                         So, as suggested by Theorem 3 the product of upper triangular matrices is in fact an upper triangular matrix. [Return to Problems]   (b)   Here’s the work for finding .                                                                       So, again as suggested by Theorem 3 the inverse of a lower triangular matrix is also a lower triangular matrix. [Return to Problems]   (c)   Here’s the transpose and the product.                                                                                So, as suggested by Theorem 5 this product is symmetric even though D was not symmetric (or square for that matter). [Return to Problems]   (d)   Here is the work for finding .                                    So, the inverse is                                                              and as suggested by Theorem 5 the inverse is symmetric. [Return to Problems]