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okay for this screencast we are going to go over two things were going to go over first eigen vectors eigen vectors so we went over how to do this for a two-by-two matrix in class Im just going to use the exact same example that I did in class so Im going to throw a clear CLC and close all so theres my matrix one negative two and negative two one I can compute the determinant I see that its negative 3 which means the inverse exists I can compute the inverse of a theres my inverse I can also do the Augmented form of the identity matrix with a and the identity matrix and I get the identity on the left and the inverse on the right with using rref and now what I want to do is I want to get my eigen values which is equal to MATLAB has a ite of a so then there are my eigenvalues negative 1 and negative 3 now in order to get the eigen vectors which you actually want to do is you want to do V
think its V comma L equals e IG a and then I ran that and over here you see here your eigen vectors negative zero point 707 one negative zero point 707 one and then over here negative zero point seven sm1 and then zero point seven two seven one so if you notice the the this V matrix is an orthonormal matrix so if I do V times V transpose I get the identity if I do V times V transpose the other way I still get the identity so it means that the inverse inverse of V is V transpose because it is ortho normal and what I also can do with these eigenvectors and eigenvalues is I can actually decompose my matrix into V and V transpose and lambda to do that I say a reconstructed is equal to V times L times V transpose and notice by the way that L when you do V comma L EIG of a you get negative 1 0 0 3 what they do is they put the eigenvalues on the diagonal like that so when I hit
f5 here notice that a reconstructed is the same thing as the a matrix that I started with if I type in a I get the exact same thing theres a little bit of numerical precision error as always but its still largely the same so in MATLAB doing eigenvalues and eigenvectors is extremely easy now what you can do is this helps you with the inverse because there are multiple ways to solve for the inverse so I kind of did this too quickly but remember when we solve systems of equations you simply do re f of a augmented with B where B is the right-hand side of the equation well it turns out you can actually get the inverse of a by augmenting it with the identity matrix and then doing row reduction so ie ye is just the identity matrix so heres I the identity matrix and then if I augment a with the identity matrix and then perform row reduction I get the inverse on the right and MATLAB also has an a built-in function to compute the inverse doing inverse of a
so I get the same thing here as I do here and then finally I can do another way where I can actually say a in I can use this guy here so I can actually do V times 1 over L times V transpose and notice that I get the exact same thing so there are multiple different ways this is using row reduction this is using built-in MATLAB functions and this is decomposing the matrix into eigen vectors and eigen values and computing the inverse now what the inverse actually is is if I take a times inverse a and for some reason it didnt like it so it worked and then it didnt work hmm okay I think the reason why I doesnt like this is because L whoops this is L negative one zero zero what we want is l equals 1 over negative 1 0 0 1 over 3 thats what we want so let me see if I can figure out how to do that real fast okay so the best way to do this is if I take L L is
negative 1 0 3 if I do L raised to the minus 1 I get what I want where I invert both of these so what I want is actually L raised to the minus 1 and that should be it ok there we go so basically I have the identity matrix is here 1 0 0 1 I have the Augmented matrix identity or sorry well initially if the a matrix and then identity and then I wrote it and I get identity on the left then the inverse on the right I use the built in MATLAB function I get the inverse and then I use item vector I get value decomposition to get vlv transpose and the inverse is just VL to the minus 1 v transpose and if you look at L and L to the minus 1 all you do is just do L equals so I L inverse equals minus 1 over minus 1 0 0 1 over 3 and that gives you your L inverse and thats all I have for right now hopefully that will help you out good luck
Inverse, MATLAB, Linear Algebra (Field Of Study), Eigenvalues, Eigenvectors, Eigenvalues And Eigenvectors, Inverse Element
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