CHAPTER 01.06: INTRODUCTION: What is an upper triangular matrix?

In this segment we will talk about what an upper triangular matrix is; a matrix of size N by N. So for example, order by N x N, which means it is a square matrix. If you find out that for which AIJ is equal to 0 for all elements in which I is greater than J, then it is an upper triangular matrix.

So the first thing to realize is that matrix has to be a square matrix. The second thing is that the following elements have to be 0, where you would find out the row number is strictly bigger than the column number. The value of the element in the matrix has to be 0. This will be clear once you look at an example.

So let’s see, we have 2, 3, -6, 5, 3, -7, 0, 0 and 0. So what you are finding out is that - let’s suppose this is an upper triangular matrix. It is a square matrix and you are finding out here are the 0’s in the upper triangular matrix. If you look at this element, this is 2nd row 1st column. This is 3rd row 1st column. This is 3rd row 2nd column. And you see something common in these 0 elements, that the row number is always bigger than the column number. That’s what we mean by I being greater than or equal to J.

So it is only where the 0’s are which dictate if a particular matrix is upper triangular. Many times students think that it depends on which elements are non-zero. Any of the elements that are on the diagonal matrix or are above the diagonal can be 0 or non-zero. But a particular matrix is considered to be upper triangular based on that fact any element for which the row number is bigger than the column number has to be 0. Let’s take another example here; B is equal to - I’ll put the same matrix except for one small difference - I’ll say 0, 5, and 3 and I’ll say 0, 0 and 0, for example. The only difference or change I’ve made is this element right here. I made it a 0. And many times people/students will think if an element is 0 on the diagonal or above the diagonal, then it is not upper triangle. But upper triangular, the nature of the matrix is defined, not by what is non-zero but by what is 0, which is below the diagonal. Since this element, this element, and this element are 0, then it is an upper triangular matrix, irrespective of this element right here. And that’s the end of this segment.