In this quick video, I'd like to clarify the differences between tensor product and Kronecker products which have used in the previous video, but I haven't nicely mentioned by name.
Both products are denoted by the same symbol which is the circled-times symbol, ⊗, and guessing it probably little bit confusing.
[미궁속을 헤메일때 한줄기 구원의 빛이 있으니 바로 요약 이더라...]
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The circled times symbol has number of different meaning in math and what I'm gonna discuss in this video will be Tensor product which is operation on tensors and Kronecker product which is operating on arrays.
And the basic summary is the tensor product combines two tensors into a new 3rd tensor, and the Kronecker product combines two arrays into a new 3rd array.
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Start with 'Tensor Product'. So just review we write the basis for vector space V as these vectors, e_1 and e_2. The basis for the Dual Space V* are these covectors, ε_1, ε_2.
The covectors are linear function that was defined by these rules here or the covector ε_i acting on the vector e_j gives the Kronecker Delta as result. It just means that we get 1 if i and j is same and we get 0 if it's different.
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So early I said the tensor product takes two tensors and produces a new tensor. So, on this case we have a vector e_i and covector ε_j, and these both are tensors. We're going to combine together using the tensor product and result is e_i circled times ε_j, e_i⊗ε_j and this thing also a tensor which happen to be a Linear Maps.
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To convince this is 'Linear Maps', we can just try passing on input vector v.
So to get the output, we can just pass this vector v to the covector ε^j, and we expand vector v as a linear combination of basis vectors e_k. Then, we bring the components v^k outside since covectors are linear functions. And we can scale before and scale after. The covector ε^j acting on vector e_k becomes Kronecker delta by definition. And by Kronecker delta index cancellation rule we can cancel out the k index and get j. So, we get the vector (v^k)(e_j) as output.
So, this tensor product(or Linear Map) is really function takes vector input and produce a vector output. So, we really did construct a linear map using a vector and a covector. That's what the tensor product does. The circled-times takes two tensors and combine them to create new 3rd type of tensor.
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Now for the Kronecker product. The Kronecker product will take these two arrays which happen to be a column vector and row vector in this case. And they will produce 3rd array. So does that by taking the 1st array on the left and distribute it to every elements in the array on the right. We get this and multiply the α coefficients and we get this which is basically a row of columns.
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If we could take this, Kronecker product but another column vector, we do the same thing we just distribute this array into every elements of the other array on the right. So we get this.
And again multiplying ω, we get this which is basically a column of rows of columns. That's how the Kronecker product works. We just distribute the array on the left into the array on the right.
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Now the tensor product and the Kronecker product are technically different, but they also highly related each other. I'll show you why here.
Let's say we have the tensor product between vector v and covector α. So, this whole thing is new tensor which is really linear map.
And we can expand v and α into linear combination of basis vectors and bring out the coefficients in front to get this.
So, here we have linear combinations of basis tensors or in this case basis linear map.
And this basis linear maps are very like those special matrices that are 0s everywhere except for single entry which is 1. So these (v^i)(α_j) would make coefficients here like the entries of matrix.
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Now if we talked the Kronecker product of the column vector associated with v and the row vector associated with α, this will give us a row of columns or basically a matrix. And remember that I said that these coefficients here I'd like the entries of matrix. Well here that makes matrix right here.
So really the tensor product and the Kronecker product are basically doing the same kind of things. it's just that tensor product is combining the abstract vector and the abstract covector in the land of algebraic symbol and Kronecker product is combinign the vector array and covector array in the land of arrays.
But the components that we get from the tensor product are just component of the matrix we get from the Kronecker product. So really the tensor product and the Kronecker product are sort of the same operation. They are just doing the works in different context.
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[이전]12. Bilinear Forms are Covector-Covector Pairs
[다음]14. Tensors are General Vector-Covector Combination
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[구구단만 알아도 '텐서']
-1. 동기(Motivation)
0. 텐서의 정의(Tensor Definition)
1. 정역방향 변환(Forward and Backward Transformation)
2 벡터의 정의(Vector Definition)
3. 벡터 변환 규칙(Vector Transformation Rules)
4. 여벡터 란?(What's a Covector?)
5. 여벡터 성분(Covector components)
6. 여벡터 변환 규칙(Covector Transformation Rules)
7. 선형 사상(Linear Maps)
8. 선형사상 변환규칙(Linear Map Transformation Rules)
9. 측량 텐서(Metric Tensor)
10. 쌍선형 형식(Bilinear Form)
11. 선형사상은 벡터-여벡터의 짝(Linear-Maps are Vector-Covector Pair)
12. 쌍선형 형식은 여벡터-여벡터 짝(Bilinear Forms are Covector-Covector Pairs)
13. 텐서 곱 vs. 크로네커 곱(Tensor Product vs. Kronecker Product)
14. 텐서는 벡터-여벡터 조합의 일반형(Tensors are a general vector-covector combinations)
15. 텐서 곱 공간(Tensor Product Spaces)
16. 색인 올림과 내림(Raising/Lowering Indexes)
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