Is Second rank tensor always tensor product of two vectors?

In summary, it is not always possible to express a second rank tensor as the tensor product of two vectors, and even if it is possible, the two vectors are not unique. To find the vectors, you can decompose the tensor into symmetric and antisymmetric parts.
  • #1
arpon
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Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions:
1. Are those two vectors ##A_i## and ##B_j## unique?
2. How to find out ##A_i## and ##B_j##
3. As ##A_i## and ##B_j## has ##3+3 = 6## components in total (say, in 3-dimension), it turns out that we need only ##6## quantities to represent the ##9## components of the tensor ##T_{ij}##. Is that correct?
 
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  • #2
arpon said:
Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ?
No, it is a sum of such products.
If so, then I have a few more questions:
1. Are those two vectors ##A_i## and ##B_j## unique?
No. Even for dyadics ##A_i \otimes B_j## you always have ##A_i \otimes B_j = c \cdot A_i \otimes \frac{1}{c}B_j## for any scalar ##c \neq 0##.
2. How to find out ##A_i## and ##B_j##
##T_{ij}## is basically any matrix and ##A_i \otimes B_j## a matrix of rank ##1##. So write your matrix as a sum of rank-##1## matrices and you have a presentation.
3. As ##A_i## and ##B_j## has ##3+3 = 6## components in total (say, in 3-dimension), it turns out that we need only ##6## quantities to represent the ##9## components of the tensor ##T_{ij}##. Is that correct?
No. See the rank explanation above.
 
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  • #3
arpon said:
3. As ##A_i## and ##B_j## has ##3+3 = 6## components in total (say, in 3-dimension), it turns out that we need only ##6## quantities to represent the ##9## components of the tensor ##T_{ij}##. Is that correct?
To add to Fresh's answer: this last remark already should make you suspicious. An arbitrary (!) second-rank tensor in three dimensions has 3*3=9 components, but two vectors have 2*3=6 components. What you can do, is to decompose a second rank tensor like [itex]T_{ij}[/itex] as

$$T_{ij} = T_{[ij]} + T_{(ij)} $$

where [ij] stands for antisymmetrization, whereas (ij) stands for symmetrization. Both parts transform independently under coordinate transfo's. The antisymmetric part has 3 independent components, whereas the symmetric part has 6 components. You can even go further in this decomposition, because the trace of the tensor components also does not change under a coordinate transformation.
 
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1. What is a second rank tensor?

A second rank tensor is a mathematical object that represents a linear transformation between two vectors. It is typically represented as a 2-dimensional array of numbers and can be used to describe physical quantities such as stress, strain, and electric fields.

2. How is a second rank tensor different from a first rank tensor?

A first rank tensor, also known as a vector, has only one index and represents a quantity with magnitude and direction. A second rank tensor, on the other hand, has two indices and represents a transformation between two vectors. It is more complex than a first rank tensor and requires two vectors to fully describe it.

3. Can a second rank tensor be expressed as a tensor product of two vectors?

Yes, a second rank tensor can be expressed as a tensor product of two vectors. However, not all second rank tensors can be written in this form. A tensor product of two vectors is a type of mathematical operation that combines two vectors to create a new object, which in this case is a second rank tensor.

4. What are the properties of a tensor product?

A tensor product of two vectors has the following properties:

  • It is a bilinear operation, meaning it is linear in both of its arguments.
  • It is associative, meaning the order of the vectors does not matter.
  • It is distributive, meaning it can be distributed over addition.

5. Are there any real-world applications of second rank tensors?

Yes, second rank tensors have many real-world applications in fields such as physics, engineering, and computer graphics. They are used to describe physical quantities like stress and strain in materials, as well as to model and manipulate 3D objects in computer graphics. They are also used in tensor calculus, a branch of mathematics that deals with the manipulation of tensors.

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