Elementary Linear Algebra For University of Lethbridge Math 1410

This section introduces the transpose of a matrix: a simple, but useful operation. We jump right in with a definition.

If we write \(A=[a_{ij}]\) to emphasize the entries of \(A\text{,}\) then the transpose of \(A\) is the matrix \(A^T = [a^T_{ij}]\) where \(a^T_{ij} = a_{ji}\text{;}\) that is, the \((i,j)\)-entry of \(A^T\) is the \((j,i)\)-entry of \(A\text{.}\) Examples will make this definition clear.

It is probably pretty clear why we call those entries “the diagonal.” Here is the formal definition.

Make note of the definitions of diagonal and triangular matrices. We specify that a diagonal matrix must be square, but triangular matrices don’t have to be. (“Most” of the time, however, the ones we study are.) Also, as we mentioned before in the example, by definition a diagonal matrix is also both upper and lower triangular. Finally, notice that by definition, the transpose of an upper triangular matrix is a lower triangular matrix, and vice-versa.

There are many questions to probe concerning the transpose operations. The first set of questions we’ll investigate involve the matrix arithmetic we learned from last chapter. We do this investigation by way of examples, and then summarize what we have learned at the end.

It looks like “the sum of the transposes is the transpose of the sum.” (This is kind of fun to say, especially when said fast. Regardless of how fast we say it, we should think about this statement. The “is” represents “equals.” The stuff before “is” equals the stuff afterwards.) This should lead us to wonder how the transpose works with multiplication.

We have one more arithmetic operation to look at: the inverse.

It seems that “the inverse of the transpose is the transpose of the inverse.” (Again, we should think about this statement. The part before “is” states that we take the transpose of a matrix, then find the inverse. The part after “is” states that we find the inverse of the matrix, then take the transpose. Since these two statements are linked by an “is,” they are equal.)

We have just looked at some examples of how the transpose operation interacts with matrix arithmetic operations. (These examples don’t prove anything, other than it worked in specific examples.) We now give a theorem that tells us that what we saw wasn’t a coincidence, but rather is always true.

We included in the theorem two ideas we didn’t discuss already. First, that \((kA)^T = kA^T\text{.}\) This is probably obvious. It doesn’t matter when you multiply a matrix by a scalar when dealing with transposes.

The second “new” item is that \((A^T)^T = A\text{.}\) That is, if we take the transpose of a matrix, then take its transpose again, what do we have? The original matrix.

Now that we know some properties of the transpose operation, we are tempted to play around with it and see what happens. For instance, if \(A\) is an \(m\times n\) matrix, we know that \(A^T\) is an \(n\times m\) matrix. So no matter what matrix \(A\) we start with, we can always perform the multiplication \(AA^T\) (and also \(A^TA\)) and the result is a square matrix!

Another thing to ask ourselves as we “play around” with the transpose: suppose \(A\) is a square matrix. Is there anything special about \(A+A^T\text{?}\) The following example has us try out these ideas.

We’ll formally define this in a moment, but a matrix that is equal to its transpose is called symmetric.

Look at the next part of the example; what do we notice about \(A+A^T\text{?}\) We should see that it, too, is symmetric. Finally, consider the last part of the example: do we notice anything about \(A-A^T\text{?}\)

We should immediately notice that it is not symmetric, although it does seem “close.” Instead of it being equal to its transpose, we notice that this matrix is the opposite of its transpose. We call this type of matrix skew symmetric. (Some mathematicians use the term antisymmetric.) We formally define these matrices here.

Note that in order for a matrix to be either symmetric or skew symmetric, it must be square.

We have just proved that no matter what matrix \(A\) we start with, the matrix \(AA^T\) will be symmetric. Nothing in our string of equalities even demanded that \(A\) be a square matrix; it is always true.

So we took the transpose of \(A - A^T\) and we got \(-(A-A^T)\text{;}\) this is the definition of being skew symmetric.

We’ll take what we learned from Example 6.1.10 and put it in a box. (We’ve already proved most of this is true; the rest we leave to solve in the Exercises.)

Why do we care about the transpose of a matrix? Why do we care about symmetric matrices?

The second reason we care about them is that they are very useful and important in various areas of mathematics. The transpose of a matrix turns out to be an important operation; symmetric matrices have many nice properties that make solving certain types of problems possible.

Most of this text focuses on the preliminaries of matrix algebra, and the actual uses are beyond our current scope. One easy to describe example is curve fitting. Suppose we are given a large set of data points that, when plotted, look roughly quadratic. How do we find the quadratic that “best fits” this data? The solution can be found using matrix algebra, and specifically a matrix called the pseudoinverse. If \(A\) is a matrix, the pseudoinverse of \(A\) is the matrix \(A^\dagger = (A^TA)^{-1}A^T\) (assuming that the inverse exists). We aren’t going to worry about what all the above means; just notice that it has a cool sounding name and the transpose appears twice.

In the next section we’ll learn about the trace, another operation that can be performed on a matrix that is relatively simple to compute but can lead to some deep results.