What is a Tensor Product and How Does it Relate to Vectors and Matrices?

Click For Summary
SUMMARY

The discussion centers on the concept of the tensor product and its application in transformations involving vectors and matrices. The transformation T is defined as T = Σ[(1/di)ai X ai], where ai are mutually orthogonal directions and di are the dimensions of a rectangular plane. The user questions the symmetry of the resulting matrix T after expansion, specifically the elements T11, T12, T21, and T22. The confusion arises from the expectation of symmetry in the tensor product, which is clarified through the properties of the involved vectors and dimensions.

PREREQUISITES
  • Understanding of tensor products in linear algebra
  • Familiarity with vector spaces and orthogonal vectors
  • Knowledge of matrix operations and properties
  • Basic concepts of transformations in geometry
NEXT STEPS
  • Study the properties of tensor products in detail
  • Learn about the implications of symmetry in matrices
  • Explore examples of transformations using tensor products
  • Investigate applications of tensor products in physics and engineering
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of tensor products and their applications in transformations involving vectors and matrices.

ngc_1729
Messages
4
Reaction score
0
Hi, can anyone please explain me how to understand this term? I tried to expand it, but seems I may not be right, so can anyone help me with expasion of this rhs term below? T is suppsoed to be symmetric, but when I expand it it doesn't seem to be symmetric, please help.

consider 2 mutually orthogonal directions a1,a2. associated with sides of a rectangular plane whose sides are d1,d2. and this rectangular plane is oriented at an arbitrary angle wrt global x axis.
Now consier a Transformation T as a function of (a1,a2) and (d1,d2) as :
T = \Sigma[(1/di)ai X ai] where i =1 to 2 and X is tensor product

when I expanded rhs of the above experssion I got:

T11 = a1 d1/d1 , T12 = a1d2/d1, T21 = a2d1/d2 , T22 = a2d2/d2
am I correct? if I am why is this not symmetric?
 
Last edited:
Physics news on Phys.org

Similar threads

Undergrad What is the difference between a complete basis and an overcomplete dictionary?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate What is the Tensor Product of Vectors and How Does It Differ Across Contexts?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad How do Tensors "work" in relation to linear algebraic objects?
  • · Replies 7 ·
Replies
7
Views
1K
Graduate Can Einstein Tensor be the Product of Two 4-Vectors?
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Dyad as a product of two vectors: what does ii or jj mean?
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Expressing matrices as outer product of two vectors
  • · Replies 12 ·
Replies
12
Views
17K
Graduate How is the Tensor Product of Vector Spaces Constructed?
  • · Replies 12 ·
Replies
12
Views
9K
Graduate Product of 3rd rank tensor with squared vector
  • · Replies 9 ·
Replies
9
Views
4K
Graduate Symmetric vector to tensor representation?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Divergence of product of tensor and vector
  • · Replies 18 ·
Replies
18
Views
18K