The Kronecker Product of two matrices - an introduction
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- Опубликовано: 30 сен 2024
- This video explains what is meant by the Kronecker Product of two matrices, and discusses some of this operation's uses in econometrics.
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Great video man! Just at 2:00, I think you meant the number of rows of the second matrix has to be the same as the number of columns of the first matrix.
After several hours of me squinting through my lecture notes for the definition of the kronecker product, I find the answer here in less than a minute. Thanks!
literally eh
Danke Herr Lambert!
I really appreciate you helping us from the scratch especially because I have a condescending professor who says we are supposed to know everything and he won't waste time clearing up stuff!
Very helpful, in less than two minutes, I was able to get what I needed. Thank you.
good video
this helped me build a quantum computer simulator thanks
Thank you for this amazing content, Ben!
But what does this product mean?
Thank you so much!!
Thanks for the video.. your voice dropped at time 1:12 of this this video when you explaining about certain application and giving us an example.. What did you say as the example ? Trying to find the intuition behind this product and that example may help.
Thanks for this video, very helpful made a lot more sense than my lecture notes writing it out in summation form.
Minor critique: your voice came through a wee bit quiet on my side, maybe that's just me though.
Thanks again for the educational content though!
1:54 should be "the number of columns of the 1st matrix must be equal to the number of rows of the 2nd matrix."
Good solution ❤
2022.10.13 thanks.
Thank you very much for this introduction. Now I can say that I understand !!
Thank you, sir. It was really helpful for me.
That was a really helpful video. Thank you!
Thank you. It's really very helpful.
thank you, this video is really helpful☺
if two matrices a and b are given find out :
1) (a kronecker b)
2) using elements of (a kronecker b ) determine (b kronecker a)
hey could you provide a solution for this problem? could you also give an example for it ?
hallo great video! one question: if i was given the A(.)B could i extract matrix A or B
Hi, in general no. There are an infinite number of products that result in the same overall result. Hope that helps, Ben
well thats even true for numbers . but specificlly for prime numbers you can! can we define prime matrices ? that would be intresting
CartanSubalgebra cute idea...
Vector spaces are always defined over a field. Thus the equation a*b=c with fixed c still has all possible solutions.
Great vedio. Well explained
Really helpful, Thankyou
Nice, concise and to the point
Very well explained sir
God Bless You Ben
Thanks sir
DUDE!!!! THANK YOU!!!!!!!!!!!!!!!
Right
y tho?
The Kronecker product essentially is used to combine matrices. In fact, one useful application is in statistics.
Imagine I have two 1x2 matrices, [1, 0] and [0, 1]. Let's give an interpretation to what these values mean.
Imagine we're going to flip a rigged coin. Let's say that for these 1x2 matrices, the first value represents "the probability of our coin landing on heads" and the second value was "the probability of our coin landing on tales". We can call the first probability p(H) and the second p(T), such that our matrix is [ p(H), p(T) ].
So the first matrix says there's a 100% chance of heads and a 0% chance of tails, the second matrix says there's a 100% chance of tails and a 0% chance of heads.
What happens if I take the Kronecker product of these two matrices?
Let's say our matrices are [a1, a2] and [b1, b2]. Then it's obvious we get [a1*b, a2*b] which would expand to [1 * [0, 1], 0*[0, 1]] or [0, 1, 0, 0].
So, we have our matrices [1, 0] and [0, 1] and doing the Kronecker product gave us [0, 1, 0, 0]. How can we interpret these results?
Since we had two matrices, you can think of this as flipping two coins. When we flip two coins, we now have four possible outcomes: both landing on heads, both landing on tails, the first landing on heads and the second on tails, and vice-versa.
So this means our second matrix represents these four outcomes: [ p(HH), p(HT), p(TH), p(TT) ].
In this case, our matrix is [0, 1, 0, 0], meaning we have a 100% chance of our outcome being the first coin landing on heads and the second landing on tales (HT).
Why is this our results? Because for the two matrices we combined, the first said p(H) = 1 and the second said p(T) = 1, so the only possible outcome is p(HT) = 1, which is what the Kronecker product is showing us.
This also shows why order is important with the Kronecker product (it is not associative). This is because if I combined the matrices in the other order, [0, 1] then [1, 0], I would get the result [0, 0, 1, 0], meaning there's a 100% of p(TH) rather than p(HT), because I flipped the coins in a different order so my outcome order is going to change.
A fair coin toss would give us the matrix [0.5, 0.5], since there's a 50% chance of a heads and a 50% chance of tails. If we flipped two fair coins, we'd get this:
[ a1*b, a2*b] = [0.5*[0.5, 0.5], 0.5*[0.5, 0.5]] = [0.25, 0.25, 0.25, 0.25]
This shows us that p(HH) = p(HT) = p(TH) = p(TT) = 0.25. Which is exactly what you'd expect from a fair coin toss.
This is effectively one way you can intuitively think about what the Kronecker product is doing.
Its a year later and no one upvoted you, but if you're still around thank you for the explanation, you explained this very well! Excellent comment@@amihartz
@@amihartz Great exposition. Did you mean commutative instead of associative? OR is the Kronecker product not commutative and also not associative?