How to multiply matrices of ANY size -- the Kronecker product.
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- Опубликовано: 16 дек 2022
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If anyone is interested in some applications of this, it is incredibly important in quantum mechanics, specifically quantum entanglement. For example, if we have a state consisting of two entangled qubits A and B (living in Hilbert spaces H_A and H_B respectively), then we can express the state as a tensor product of A and B. Then we can calculate various objects of interest (e.g. expectation) to find out information about the state.
Doesn't entanglement mean that a composite state of A and B can't be written as a Kronecker product?
I seem to remember outer products and commutation relations of Pauli matrices (when I studied this over a decade ago) - which feels like the same sort of idea...
Yep, my first thought when I saw this was "quantum state vectors". It's also very important in quantum field theory.
It's weird that I'm *both* intimately familiar and completely ignorant of this matrix product; in the sense that I use it all the time in physics, but I never think of it as a matrix product :-)
The way I handle tensor products in physics is by the simple rule "each operator 'hits' the vectors in its own space". That, and operator linearity, is sufficient to disambiguate any tangled expression involving tensor products.
I remember seeing a "proper" vector tensor product as an exercise in a quantum mechanics textbook, and it took me a while to parse it. It was a product from C^2 (2-dimensional complex space) times C^2 -> C^4, and it constructed the C^4 basis vectors in a straightforward way, i.e.:
(1,0) x (1,0) -> (1,0,0,0)
(1,0) x (0,1) -> (0,1,0,0)
(0,1) x (1,0) -> (0,0,1,0)
(0,1) x (0,1) -> (0,0,0,1)
Really simple encoding, when you give it a little thought, but to me it was utter overkill; since the bases in different spaces never interact (so I reasoned), why not simply *keep* (1,0) x (1,0), etc., and act with the respective operators in the bra-ket notation? But now I appreciate the mathematician's way.
The Kronecker product is also super important for many-body quantum mechanics. For example it pops up when computing total angular momentum of a multi Fermion system (also for bosons), if J=J1+J2 then J² = J1²+J2²+2J1x⊗J2x+2J1y⊗J2y+2J1z⊗J2z.
I don’t think it’s possible to have too many videos about tensors, tensor products, etc. - so definitely do more please!
I think the tensor product video would be great! Also I would really enjoy a more abstract extension further down the line (probably in the faaar future^^) to representation theory and irreducible representations of higher dimensionality from the tensor product for goups/algebras
^^^^ yes please!
Tensor products can seem a bit intimidating, so videos going through the tensor algebra and its ramifications sounds AWESOME!
Your channel is way underrated as per the current number of subscribers. There are about a dozen other invaluable math channels out there (that I am aware of), but if I had to pick only one to keep watching, it would undoubtedly be yours. The combination of your wide but wise choice of topics and your consistently fluent delivery without resorting to teatrical gimmicks are unparalled.
Thank you so very much for your inmense contribution to the spread of highly interesting and beautiful mathematics.
I would enjoy seeing the follow-up video (tensor product of linear transformations).
More videos about tensors would be amazing! Maybe even getting into tensor calculus
Awesome!! Advanced math day is always my favorite day. Definitely want to see more about tensors, vector spaces, algebras, etc -- I'd defer to whatever you want to make, but I have also been really invested in representation theory (trying to understand the connection between the quantum zoo and representations at different "weights"/eigenvalues) and it would help greatly if you made some content on representation theory of Lie algebras/groups!!
4:42 I think that might be fun to…make😂. Michael knows what he’s talking about.
This is so linked to the tensor product that I honestly didn't know it had a distinct name; in my mind it was just "the tensor product of matrices" which of course isn't really accurate, but was good enough for what I needed (QM, mostly)
You inspire me (nearly) every day! Thank you sir!
Video about tensor product of linear transformations sounds great, I love your more abstract videos
Yes! please do more on tensor products
A few months late, but another application for the Kronecker product is the ability to factorize the Discrete Fourier Transform into factors that correspond to Fast Forier Transforms.
It also features in my favorite Lemma involving the vec operator: vec(ABC)= (C' kron A) vec(B) which is incredibly useful in matrix diferential calculus.
12:08
Feels like some sort of representation of an outer product (which would be 4 dimensional) in 2D. Almost using the array of matrices for the first two indexed dimensions, and then locating within each matrix to pick up the extra two dimensions. Rather like a 3D image may be built up of a 1D (e.g. vertical axis) list of 2D cross sections, we have 2D cross sections which are "layered up" in a 2D manner.
Good stuff!
Tensor products would be fascinating, your videos are brilliant
I would also like to see the Kronecker product as a matrix representation of linear transformations.
I just used the Kronecker product last week, to produce the matrices for the transverse Ising model
I'd watch a video on the tensor product. And great video!
That was in my linear algebra qualify to show its mixed multiplication property, inverse property follows naturally after, and then show how the eigenvalues and eigenvectors relate to those of A and B
Also, still waiting on the followup from the other tensor video, about relating tensors/notations among disciplines of STEM! (relativity w/ einstein notation, solid mechanics/engineering, pure math, etc). I hope there's enough engagement for it, but regardless that's the kind of content I subscribed for however many years ago and stay excited for!
Wow the t-shirt. The lighting turns it Klein blue. But it's also very good on the unlit side.
Graph tensor product is also defined in terms of kronecker product of adjacency matrices.
You can represent any topology on a finite set as a unique reflexive and transitive graph.
Guess what topological product of n such spaces can be represented as kroenecker product of adjecency matrices of those spaces.
It also really plays nicely with vec : R^(nxn) -> R^(n*n) function
Unlike matrix multiplication, Kronecker and Hadamard products preserve positive definiteness, which makes them useful for Gaussian processes.
yes, i would like to see a tensor product video. thanks
I want to see a whole video about the tensor product
Michal penn thank-you so much, our physics professors uses them but never clarify why they are true. You are blessing for us who what to actuallyunderstand physics and maths. And please video about mathematics in particle physics are always welcome.
Can you discuss the Jordan Canonical Form Michael?
It's a Matrix of Matrices. That doesn't really look like a product, though it has some conceptual similarity with a cartesian product of sets.
Having said that, the Kronecker product is important in quantum mechanics, so whatever term we use for it we physicists have borrowed it for our practical uses
The determinant is a homomorphism between the set of invertible matrices and the set of non zero real numbers
Hey Michael, I've got a very nice problem for you :
Find a closed form of the simple infinite continued fraction with 3 parameters
u(a,b,c) = a + 1/(b+(1/(b+(1/...+1/(b+1/(2a+1/(b+1/(b+1/...+1/(b+1/(2a...)))))))))) where in every period (between 2a and the next 2a) the number b appears c times. This can be written in compact continued fraction notation as
____________
u(a,b,c)= [a;b,b,b,...,b,2a]
\ /
\ /
"b" appears c times here
I should point out that here we assume that a, b and c are positives integers, and u(a,b,c) converges.
Hint: try exploring with some fixed values for the number c, and then build some sort of recursion based on c :)
I think this is one of my greatest results and I'd be very happy to see you cover this
What is the tensor product?
Amazing video...
Already 0:40 this looks like multiplying A by B where B is treated as the scalar field of A
This feels like a cheat code
Tensor products are my Achilles heel. Maybe I am too relaxed? But seriously I would very much like to see you do more on tensor products.
Kronecker products are used in time series econometrics
The only part that I am confused about is the difference, if any, between the Kronecker product, tensor product, and the direct product.
we can prof d fact by polynominal
better question, *why* to "multiply" (=combine)
Too blue
More stuff like this instead of "an interesting problem from the 1969 Outer Mongolian Math Olympiad"
no more video like above pl.
lols