Diagonalize 2x2 matrix

alright thanks for watching and in this video I want to show you how to diagonalize a 2 by 2 matrix for example if diagonalize the matrix 7 2 - 4 1 and of course if I can do it so can you so let's find your eigen values so suppose this is a and to do that all we have to do is calculate determinant of lambda I minus a by the way for all the geums out here this is for you because this is Li minus a so Leon with or other the odds also have there by the way I calculated lambda I minus a to get rid of sign mistakes at the end but it's totally fine to also do a minus lambda I so all the leaves over there Gulin cuz you wanna set them to 0 so let's do determinant of lambda I minus a which means on the diagonals you put lambdas and everything in the matrix you put minus so minus 7 minus 2 4 and minus 1 and again seems paradoxical why would I do all the minuses here it makes the end of it easier okay and now let's calculate the determinant so it's lambda minus 7 lambda minus 1 and then plus minus 4 times minus 2 and it's weird cuz it looks almost factored out but the thing with eigenvalues is best to expand every thing out here so let's foil it out lambda squared minus eight lambda plus seven plus eight and that's lambda squared minus eight lambda plus fifteen and let's factor this out because you want to set it equal to zero and I believe in this case it becomes I think lambda minus three times lambda minus five because the product is fifteen and the sum is -8 and 4 I get values we want to set this equal to zero which tells you that either lambda equals to three or lambda equals to 5 so we have effectively found the eigenvalues and the next step is for each eigenvalue you found find a null space in other words find the eigenvectors in fact let's keep this that was lambda I minus a and by the way the reason I left it in this form is because you had like lambda squares turning around if used a minus lambda I you would have lots of - lambdas and you're at that point you might be too tired and make a sign mistake okay now I get for every eigenvalue you want to find null space of lambda I minus a so find in this case no space of three I minus a it looks like you have to recalculate stuff but no because remember that was lambda I minus a so all you need to figure out is to plug in lambda equal to 3 so that's no space of 3 minus 7 minus 2 4 + 3 minus 1 which becomes null space of minus 4 minus 2 4 - yay 42 and then you just do one more row reduction you do one more row reduction so if you really read it will reduce this if you want divide both things by 2 or minus 2 so 2 1 2 1 and then this row reduces so 2 1 0 0 and how do you find a null space you just solve the equation ax equals to 0 so 2 1 0 0 XY equals to 0 0 and again this is what you want to find gives you 2x plus y equals to 0 and 0 equals to 0 which is not very interesting and then this gives you y equals to minus 2x and remember you want to find X Y so this so X y equals 2x and minus 2x but this is the same as x times 1 minus 2 and because X was arbitrary this tells you that the null space is just the span no space of I guess a a 3 are sorry 3i minus a it's just equal to the span of 1 minus 2 in other words the eigenvectors corresponding to 3 is just 1 minus 2 in just two remarks in order first of all this is never 0 if you find that you find the null space is 0 then you definitely made an algebra mistake either you found the wrong I can value or you made a mistake in your row reduction because by definition of eigen vector you should always it's not the span is not 0 so an eigen vector corresponding to three I minus a an eigen vector corresponding to lambda equals to 3 is really a vector such that this null space is nonzero and second of all it's normal that two eigen vectors are always in the form of span of something because if v is an eigenvector then so is any multiple always forms a vector space again corresponding to 1 value okay let's just do the same spiel but with lambda equals to five so the null space of you know 5i minus a well that's the null space in this case you just plug in lambda equals to five so remember lambda I minus a was lambda minus seven minus two for lambda minus one plug in lambda equals to 5 you get 5 minus 7 minus 2 for five minus 1 and that's the null space of minus 2 minus 2 for 4 how nice you can simplify this so that's the null space of 1 1 1 1 and it's the null space of 1 1 0 0 and now let's solve the equation this XY equals to 0 0 that just means X plus y equals to 0 so y equals 2 minus X and therefore x y equals 2x minus x those a kiss again which is x times 1 minus 1 and therefore what this tells us is the null space it's just a span of 1 minus 1 in other words an eigen vector corresponding to lambda equals to 5 is 1 minus 1 so great we found the eigenvalues we found the eigenvectors and the question is how do you write down sir so usually mini algebra problems they don't say hey find the eigenvalues find the eigenvectors what this say is fine matrices a and P such that so fine it's a DNP where a equals to PD P inverse and in another video I'll give you a cool Legend of Zelda analogy explaining why this means that a is like T or a similar to D so basically fine matrices DNP such as a is similar to D but what are the requirements well p is invertible and d as it says it's diagonal and this is why it's called diagonalization we want to turn a into a diagonal matrix and here's a cool answer d turns out to just be the matrix of eigenvalues so since it's diagonal here we have 0 0 and you put the eigenvalues here 3 and 5 and 4 p you just put the corresponding eigenvectors and here the order is important if you put 3 here you have to put the eigenvector with 3 which in this is is 1 minus 2 and if you put 5 in the second column the second eigenvector has to be 1 minus 1 and there you go you find you you know successfully diagonalizes me and there are many many many applications of diagonalization I invite you to look at some of them and other linear algebra videos to see how cool that is alright so if you really like this linear algebra extravaganza and want to see more videos please make sure to subscribe to my channel thank you very much

Channel: Dr Peyam

Views: 24,300

Rating: 4.927711 out of 5

Keywords: diagonalization, diagonalize, eigenvalue, eigenvector, eigenvalues, 2x2, matrix, linear algebra, math, lay, chapter 5

Id: H-NxPABQlxI

Length: 12min 22sec (742 seconds)

Published: Thu Jul 19 2018