 All right, everyone. We're in the last part of section 6.1 for the textbook linear algebra done openly. And I wanted to talk about eigenvalues with respect to triangular matrices. It turns out this idea of triangular matrices as it's related to eigenvalues is gonna be connected to determinants, which actually is sort of like, there's a nice connection between eigenvalues and determinants that we're gonna talk about here. So it turns out that the eigenvalues of a triangular matrix are just the entries of its main diagonal. So for a triangular matrix, there is no calculation you have to do to find eigenvalues. It's really nice. So I wanna give you a quick argument why that is, right? So let A be a triangular matrix. And if it's a triangular matrix, then what it's gonna look like will be something of the following form. Your matrix A would look like, we'll just assume it's upper triangular or a lower triangular matrix. We do something similar. We could take some entries. We're gonna call it lambda one is the first diagonal, lambda two, lambda three, all the way down to lambda N. And all the entries above the diagonal we don't really care about. We'll just put them as stars, right? There's something, but we don't really care what they are. That's not gonna be of interest to us. And then as it's upper triangular, everything below is going to be, everything below there is gonna be zero as well. So we get something like this here. So just focus here on these diagonal entries right here. And so because the matrix is triangular, what happens when we take this matrix A, excuse me, A minus lambda I, right? And let's say we have a specific lambda I in mind. What if we subtract lambda Y I? So A minus lambda I I like this. Well, if you did that, and so like say that this was like the second position, right? If we subtracted lambda two from all of these pieces, for the most part, everything's unaffected, but I'll notice bringing your attention to this one right here is somewhat significant, right? In this situation, if you were to subtract lambda two from lambda two, kind of erase that out there, you'll end up with just a zero sitting right here. You get this zero. And so you'll notice that this matrix right here is gonna be singular because a triangular matrix is singular if and only if it has a zero along the diagonal. And so the thing is since A was triangular, A minus lambda I will be triangular as well. And this matrix here is singular if and only if it has a zero on the diagonal. And this will happen if and only if lambda is a diagonal entry. So for diagonal matrices, you get this cute little trick here that the diagonals are the only, so all the diagonal issues are eigenvalues and also nothing other than a diagonal number for these triangular matrices will be an eigenvalue. So let's actually look at two triangular matrices for a moment. The first matrix A you see, it's an upper triangular matrix and its diagonal entries are right here. So what are the eigenvalues? The eigenvalues are gonna be three, zero and two. And I wanna draw your attention to a moment right here that it is perfectly okay for zero to be an eigenvalue. We say that the zero vector is not considered an eigenvector, but the zero number can be an eigenvalue. The eigenvalues for this matrix are just gonna be zero, three, two because it's upper triangular. Example B right here, it's a lower triangular matrix but that doesn't make any difference for us. We get that the eigenvalues are looking at the diagonals four, one and four. Right? Now, admittedly this four shows up twice on the diagonal. So if you wanna list it's just the eigenvalues are four and one, that's okay. But sometimes we like to list these things when they're repeated, we wanna list them twice because the idea is that this is a repeated eigenvalue. The significance of a repeated eigenvalue will make more sense forthcoming, but for the moment this matrix has a repeated eigenvalue of four but the eigenvalues of the matrix are gonna be four and one. So that's a super simple way to find the eigenvalues of triangular matrices and it comes from this fact about the determinants basically because the determinant of a triangular matrix is just the product of the diagonals. We're gonna see in the next section that determinants and eigenvalues are connected more than what you might initially suspect. And because we've seen how to check in this section if a vector is an eigenvector, if lambda is an eigenvalue or not. And if we find an eigenvalue, we can find the eigenvectors. So the question comes down to how do we find the eigenvalues? For triangular matrices it's pretty simple but what do we do when the matrix is not triangular? We're gonna talk about this more in section 6.2 as we discuss the characteristic polynomial. Thanks for watching this video. Either this part or the whole series are some other parts of it. It's good to have everyone in their support. If you like what you see, feel free to like this video or all the other videos. Leave comments if you have any questions and glad to answer them. Subscribe if you like these videos and wanna see some more of them in the future. I will see you next time, stay tuned. Stay healthy everyone and have a great day. Bye.