 Hi everyone. We're in section 6.1 Eigen values and eigenvectors for our textbook linear algebra done openly and I want to ask ourselves a question How do we know if a vector is an eigenvector? The definition of an eigenvector and an eigenvalue were given the last video check that out If you're not sure what that means here But how can we check if we have an eigenvector or not? It turns out it's a fairly simple check that I want to show you in this video right here So we have a matrix a which is given to you that 2 by 2 1 6 5 2 and I have two vectors How can we check to see if these are eigenvectors or not? It turns out all we have to do is actually just multiply together the matrix and the vector So if we take a times u 1 6 5 2 is our a and then u is 6 and negative 5 So first just do the product of the matrix and the vector So we end up with 6 minus 30 for the first entry there and then we get 30 minus 10 Simplifying that what would he end up with we're gonna get negative 24 and then 20 Like so so we calculate the the matrix vector product And then we have to see it can we factor it right because if you have an eigen vector It'll satisfy the relationship that ax equals lambda x or lambda is just some number, right? So we have to see is the product we got divisible By our vector 6 and negative 5 or another way is if we look at these numbers Can we find some common factor between them so like negative 24 and 20? I just can't help myself but notice. Oh, yeah, they're all divisible by 4 So if we factor out the 4 We take out the 4 we end up with a negative 6 and 5 now. Is that the vector u? Oh, it's kind of close. The sign is off I have a I have a negative 6 when I want a 6 and I have a 5 when I have a negative when I expect a negative 5 That's actually an easy thing to fix factor out the negative one now. So we actually have negative 4 times 6 and negative 5 and so you'll now notice that this is negative 4 times the original vector u So the answer to the question is yes Yes here U is an eigen vector It's an eigen vector and then we also know it's eigen value. It's eigen value is equal to negative 4 What about the other vector a times v? Well, if we do the matrix multiplication this time, let me write down a 1652 That should be a 2 And then we times that by v which is 3 and negative 2 When we look at this product again doing the details here, we get 3 minus 12 and we're going to get 15 minus 4 Simplifying that we get negative 9 and 11 As our vector product that time well, is that is that a Multiple is this a multiple of v right? Is this a multiple of v right so we could try to sort of randomly guess Scalers to fact factor out by looking at gcds and things like we did last time But really we know what we want the vector to look like we want a 3 and a negative 2 So what do we multiply by 3 to get negative 9? Well, notice that negative 9 would have to factor as negative 3 times 3 So if this is an eigen if this is some eigen thing we'd have to pull out negative 3 Leaving behind a 3 right here But what do we do when we factor out negative 3 from 11? We get like 11 over negative 3 that is not going to simplify to be a negative 2, right? It's not negative 2 And so to answer the question here We actually would have to reply in the negative here. We would say that no v is not An eigen vector an eigen and when we say eigen vectors here Of course, we're referring to a specific matrix. It's not an eigen vector 2a and this previous thing here We also were talking about it's an eigen vector to a specific matrix a just because this isn't an eigen vector to this matrix It doesn't mean it couldn't be an eigen vector for another matrix And so it's a really nice check to see Whether a vector is an eigen vector for a specific matrix or not Just multiply the matrix and the vector together and then try to factor the product and see if you can get a multiple of the original The original vector