CHAPTER 04.01 - 04.05: PRIMER ON SIMULTANEOUS LINEAR EQUATIONS: Upper and Lower Triangular Matrices
In this segment, we're going to look at what the definition of an upper triangular and a lower triangular matrix is. So we are basically reviewing some of the things which you need to know about solving simultaneous linear equations. So upper and lower triangular matrices come into the picture when we talk about LU decomposition method of solving simultaneous linear equations, or the Gauss elimination.
So let's go ahead and see what an upper triangular matrix is. So, an n-by-n matrix, a square matrix is called upper triangular if the elements, the following elements are 0, which is aij is equal to 0 for i greater than j, so then it'll be called an upper triangular matrix. So what does this mean? So an n-by-n matrix A, I should call it, so that these . . . this is the corresponding element of that particular matrix. So an n-by-n matrix A is called an upper triangular matrix if aij is equal to 0 for i great than j. What does that mean?
So let's take an example, let's suppose I have a 4-by-4 matrix, and I write as 20, 3, 6, 0, -5, 2, 1, 3, -6, 20, and then 0, 0, 0, 0, 0, 0, here. So what you are basically seeing here is that any of the elements where the row number is bigger than the column number, they are all supposed to be 0, so those are the elements which are below the diagonal. So these are elements which are below the diagonal, and they're all 0. So keep in mind that the upper triangular matrix is based on which elements are 0, not the elements which are nonzero. So all you have to look at is if the elements below the diagonal, the diagonal is this one, where the row number and the column number is exactly the same, so any of the elements which are below that diagonal, they have to be 0. So, for example, if you are looking at this element here, this is fourth row, first column, so the row number is strictly greater than the . . . strictly greater than the column number, and you have to have a 0 there. So any of the values of places where the row number is strictly greater than the column number, it has to be 0, no exceptions. Only then you can call it an upper triangular matrix. Keep in mind that this 0, for example, here, does not make it upper triangular or not upper triangular. It is the 0s below the diagonal which make the matrix to be upper triangular or not. And so, based on that, you basically look at if the row number strictly greater than column number elements are all 0. So that's what's an upper triangular matrix.
Let's go ahead and see what a lower triangular matrix is. An n-by-n matrix, A, is lower triangular if aij is equal to 0 for i less than j, i strictly less than j.
So, let's look at an example of a 4-by-4, so we're going to take a 4-by-4 matrix here, just like we did took it previously for the upper triangular matrix, and let's suppose we have 20, -5, 3, 2, in the diagonal, and let's suppose we have 10, 0, 6, 0, 0, 3, here, and then we should have to have 0, 0, 0, here, 0, 0, 0, here, and 0, here. So, again, for determining whether a particular matrix is lower triangular, you have to look at what is above the diagonal, and if all the elements above the diagonal are 0, then you know that the matrix is a lower triangular matrix. It does not matter whether elements which are on the diagonal and below the diagonal are 0 or nonzero, what they are. What determines whether a matrix is lower triangular or not is based on what are the elements above the diagonal. Again, when we say i less than j, that means the row number is strictly less than the . . . less than the column number. So, for example, if you look at this element here, that is first row, third column, and you are finding out that the row number is strictly less than the column number, and that has to be 0. And that is the case for all the elements where this i is less than j, and that's what you are seeing here, that they are all 0, and that's what's a lower triangular matrix. And that's the end of this segment. |