CHAPTER 01.07: INTRODUCTION: What is a lower triangular matrix?

Let’s define what a lower triangular matrix is. So N by N matrix - so we are talking about a square matrix here. An N by N matrix A is lower triangular if AIJ is equal to 0 for J is greater than I. So the first thing that we have to talk about is that the lower triangular matrix has to be a square matrix; so the number of rows has to be the same as the number of columns. And then these elements have to be zero, the ones for which the column number is bigger than the row numbers. So it is based on what is 0. If the column number is bigger than the row number, that element has to be 0 for all J greater than I, for all I, J. I is going from 1 to N and J is also going from 1 to N.

Just to be clear, let’s look at an example. So let’s look at an example here for the lower triangular matrix. 6, 3, 2, 0, 0, 0, 5, 6, -7. This is a lower triangular matrix because what you are finding out is that the elements that are above the diagonal - they are all 0. So right here, right here, right here - 0. Now, let’s look at the element numbers here. So we got A 1st row, 2nd column. You got 1st row 3rd column and you got 2nd row, 3rd column. And you find out that the column number is always bigger - strictly bigger - than the row number. Column number 3 is strictly bigger than the row number 1. Column number here is 3, which is strictly bigger than the row number here which is 2. It is because of these 0’s right here, that it is considered to be a lower triangular matrix here. It does not matter what’s here on the diagonal and below the diagonal; it what is above the diagonal and those are all 0.

Many times its thought, if there is a zero somewhere here in these elements then it is not considered being a lower triangular matrix, but that’s not true. Because the lower triangular matrix is dictated by whether a particular square matrix can be considered a lower triangle or not is dictated by what is above the diagonal. What is happening in the elements in which the column number is bigger than the row number? Are they all 0? Look at another example here. We have B- I am going to use the same matrix as here but I’ll just leave one element out and change it. 6, 0, 0, -5, 0, 0, 6, -7, 2. It is the same as the matrix here except I changed this 3 to 0. But this is still a lower triangular matrix because it is these elements which determine whether the matrix is lower triangular or not. This zero doesn’t matter in terms of classifying this particular matrix as a lower triangular matrix. And that is the end of this segment.