69 - The Cayley-Hamilton theorem

another short uh short one about a specific theorem called the cayley-hamilton theorem named after two mathematicians Cayley and Hamilton and I'm not going to prove it the proof is rather subtle rather subtle but I want to set it up and then and then write it down okay so suppose we have a polynomial let's call it f of X so f of X is a polynomial a and X to the power n plus a and minus 1 X to the n minus 1 plus dot plus a 2 x squared plus a 1 X plus a 0 this is a polynomial of degree n do you agree okay so what does it mean and this is in fact a definition what does it mean to substitute a matrix into a polynomial okay it just means by definition that whenever you see X you plug in the matrix so it's just a n this is a scalar times a to the power N and we know how to raise a matrix to the power n right it's just a times a times a n times in fact if a is diagonalizable we know a shortcut of how to do it that was the theorem in the previous lesson right plus a and minus 1 a to the power n minus 1 plus dot dot a to a squared plus a 1 a plus a 0 but this doesn't compile right because these are all matrices and this is a scalar a 0 I exactly I is a to the power 0 good makes sense now so f of a is a matrix is a matrix which is precisely the matrix plugged into the polynomial may cents right okay and for an operator for an operator if T is an operator from V to V I can do the same thing I can plug it into the polynomial and what do I get I get a n T to the power n what is T to the power n by the way what what operation does this correspond to it so what's T to the power - its T times T but it's the multiplication of operators which is composition remember multiplication of operators is composition so this is T of T of T of T of T of the of V okay its composition good that's important to emphasize so plus a n minus 1 T to the power n minus 1 plus dot dot dot a squared T squared plus a 1 T plus again a zero times I where now I is the identity operator okay so plugging in an operator into a polynomial gives us again an operator good everybody so just let's write that this is here the product is a composition of operators right we know that I'm just reminding you ok so let's do a little example a short example let's take a example let's take a to be the matrix 1 2 3 2 okay the characteristic polynomial of this matrix P of alpha the characteristic polynomial is what is the determinant of a minus alpha I 1 minus alpha 2 3 2 minus alpha right and we already know we already know what this is we found it in a previous example this may be just let's write it it's 1 minus alpha times 2 minus alpha minus 6 and this equals alpha squared minus 3 alpha minus 4 ok we just did it in a previous example remember this matrix I think this was our first baby example of how to find eigenvalues and eigenvectors and and from this we we extracted the general theorem remember ok what happens when we take a and substitute it into its own characteristic polynomial okay so what is P of a so it's a squared minus 3 a minus 4 I do you agree right I substituted a into its polynomial and I get let's see what I get I get 1 2 3 2 squared some times itself minus 3 times 1 2 3 2 minus 4 I 4 0 0 4 do you agree good this equals here I do need to do something so it's 1 minus 6 negative 5 2 minus 4 negative 2 3 minus 6 negative 3 and 6 minus 4 2 what am i doing groan why don't know - you were doing a determine oh oh oh I'm so sorry sir and you're sitting there just quiet enjoying me making a fool out of myself yeah I don't know so what is it one plus six right help me out here one plus six that's a seven does everybody agree that this is a seven sorry two plus four plus six right what two plus four six everybody greed is - six my self-confidence is now completely messed up 3 plus 6 9 and 6 plus 4 10 good ok and now I hope I can multiply a matrix by three correctly 3 6 9 6 and minus 4 0 0 4 do you agree and now let's finally subtract the matrices so I have 7 minus 3 minus 4 0 6 minus 6 0 - 0 0 9 minus 9 minus 0 0 and 10 minus 6 4 minus 4 0 do you agree now I have my self-confidence back because I knew that was what I'm going to get and what how did I know what I was going to get that's the theorem so a theorem this is the cayley-hamilton theorem says the following let a bee let a ni n a n by n be a matrix with characteristic polynomial ah P of alpha then if you take a and plug it into P of alpha P of a you always get the zero matrix every matrix solves is a root of its characteristic polynomial as a polynomial in matrices okay clear I'm not going to prove it again again I'm not going to prove it the proof is somewhat a somewhat subtle and I'm gonna leave it at this as at the theoretic level okay I mean we did an example of how we calculate this sort of stuff but I'm going to leave this statement as a theoretic statement it can be used to prove other sort of statements regarding matrices and powers of matrices and and so on but we're really just kind of collecting the n details of this topic and I don't want to spend more time on on this issue so this wraps up our discussion are pretty long discussion of the whole topic of diagonalization and eigen values and eigen vectors and characteristic polynomials and so on algebraic multiplicity geometric multiplicity and everything we have one further topic remaining for this course which we're going to do very briefly very briefly and that's uh going to be next time questions okay so thank you my CT show you -

Channel: Technion

Views: 77,839

Rating: 4.8423076 out of 5

Keywords: Technion, Algebra 1M, Dr. Aviv Censor, International school of engineering

Id: eLgh1RYxViU

Length: 10min 34sec (634 seconds)

Published: Mon Nov 30 2015