@@joshuaiosevich3727 interesting thought! I'm a bit unsure where I'd start - the connection between the characteristic polynomial and eigenvalues rather than singular values might be tricky to get around. But I've been wrong plenty of times before, and I know there are all sorts of generalisations of the cayley hamilton theorem into other more exotic algebraic situations, so there could be something!
haha - in many ways you're right. I have very rarely used the Cayley-Hamilton theorem directly. But it is a great 'justifier' theorem. e.g. imagine your building a matrix that is like the controllability matrix, but bigger: [B,AB,...,A^{n-1}B,A^nB,...] How do you know when you should stop? From the perspective of lots of linear algebra properties, the Cayley-Hamilton theorem tells you there is nothing to be gained by going beyond the entry A^{n-1}B, since the rank of the matrix won't change (and so we have a complete set of independent columns, which might be enough for the types of computations we need to do, and that kind of thing). The theorem (in engineering at least) is often used this way - but when you actually do the computation, you don't use it.
I noticed that this fact is trivial if the matrix is diagonalizable.
It's always nice seeing how that kind of insight and intuition can generalise - or not. Definitely a fun part of learning. Thanks for watching!
@@richard_pates here’s a fun question, can you generalize the Hamilton Cayley theorem in general using SVD?
@@joshuaiosevich3727 interesting thought! I'm a bit unsure where I'd start - the connection between the characteristic polynomial and eigenvalues rather than singular values might be tricky to get around. But I've been wrong plenty of times before, and I know there are all sorts of generalisations of the cayley hamilton theorem into other more exotic algebraic situations, so there could be something!
I can notice by the anxiety that there are not many if any uses for this theorem. Great explanation though.
haha - in many ways you're right. I have very rarely used the Cayley-Hamilton theorem directly. But it is a great 'justifier' theorem. e.g. imagine your building a matrix that is like the controllability matrix, but bigger:
[B,AB,...,A^{n-1}B,A^nB,...]
How do you know when you should stop? From the perspective of lots of linear algebra properties, the Cayley-Hamilton theorem tells you there is nothing to be gained by going beyond the entry A^{n-1}B, since the rank of the matrix won't change (and so we have a complete set of independent columns, which might be enough for the types of computations we need to do, and that kind of thing). The theorem (in engineering at least) is often used this way - but when you actually do the computation, you don't use it.
🌷🌷🌷 thank you ^_^
thank you