The discussion revolves around the concept of tensors, exploring their definitions, interpretations, and the confusion surrounding their nature compared to vectors and matrices. Participants share various explanations and perspectives on how to understand tensors, including their mathematical properties and transformations under different coordinate systems.
Participants express a range of views on the nature of tensors, with no clear consensus reached. Some agree on the multidimensional aspect of tensors, while others contest the definitions and implications of tensor properties, particularly regarding their independence from coordinate systems.
Limitations in understanding arise from the complexity of tensor definitions and the varied terminology used in the discussion. Some participants note that traditional explanations may not adequately convey the essence of tensors.
I figure it's just a vector's bigger brother
DiracPool said:A vector's bigger brother is actually a matrix. A tensor is like the bad uncle that used to bully you and try to make you think that you were stupid because you couldn't tell the difference between co-variance and contra-variance.
Fredrik said:Let V be a finite-dimensional vector space, and V* its dual space. A tensor of type (n,m) is a multilinear map T:\underbrace{V^*\times\cdots\times V^*}_{\text{$n$ factors}}\times\underbrace{V\times\cdots\times V}_{\text{$m$ factors}}\rightarrow\mathbb R. The components of the tensor in a basis \{e_i\} for V are the numbers T^{i_1\dots i_n}{}_{j_1\dots j_m}=T(e^{i_1},\dots,e^{i_n},e_{j_1},\dots,e_{j_m}), where the e^i are members of the dual basis of \{e_i\}. The multilinearity of T ensures that the components will change in a certain way when you change the basis. The rule that describes that change is called "the tensor transformation law".
DiracPool said:A vector's bigger brother is actually a matrix. A tensor is like the bad uncle that used to bully you and try to make you think that you were stupid because you couldn't tell the difference between co-variance and contra-variance.
These are both quite a bit misleading. In a given coordinate system, we can represent a vector as a "list" of numbers and we can represent a tensor by a rectanguar grid of numbers or matrix. But the crucial point of both vectors and tensors (technically, a vector is a tensor) is that they have an existence independent of any given coordinate system. And the change as we change from one coordinate system to another is "linear homogeneous". One reason that is important is that if a tensor happens to be all 0s in one coordinate system, it is all 0s in any coordinate system. So if we have an equation, say A= B where A and B are tensors, in one coordinate system, we can write that as A- B= 0 so that A- B= 0, i.e. A= B in any coordinate system: tensor equations are independent of the coordinate system.TGlad said:My explanation is that a tensor is an n dimensional grid of numbers. Since a scalar is a 0d grid, a vector is a 1d grid, a matrix is a 2d grid, we can say that a tensor is any of these, plus 3d grids, 4d grids etc.
But it isn't just a grid of numbers, it also has a few operations on it, such as addition and multiplication, which work like matrix algebra in the 2d case, and real number algebra in the 0d case, etc.
You left out the most important thing, which is that one of these "grids" isn't enough. There must be one for each coordinate system. And the relationship between the "grids" associated with two different coordinate systems must be given by the tensor transformation law.TGlad said:My explanation is that a tensor is an n dimensional grid of numbers. Since a scalar is a 0d grid, a vector is a 1d grid, a matrix is a 2d grid, we can say that a tensor is any of these, plus 3d grids, 4d grids etc.
But it isn't just a grid of numbers, it also has a few operations on it, such as addition and multiplication, which work like matrix algebra in the 2d case, and real number algebra in the 0d case, etc.
So is calling a real a number misleading? since reals have an existence independent of coordinate system (which is just scale in 1d).But the crucial point of both vectors and tensors is that they have an existence independent of any given coordinate system.
I did mention that tensors come with a set of operators that have specific rules, which are basically like real, vector and matrix algebra in the 0d, 1d and 2d cases. I didn't suggest that they are just data structures.There must be one for each coordinate system. And the relationship between the "grids" associated with two different coordinate systems must be given by the tensor transformation law
Yes. A matrix is not a tensor. You can however define a tensor by specifying an n×n matrix, a coordinate system and the tensor type (i.e. if you want the tensor to be a map from V×V into ℝ, from V*×V into ℝ, from V×V* into ℝ, or from V*×V* into ℝ), because there's exactly one tensor of each type that has components in the chosen coordinate system that are equal to the components of the matrix.TGlad said:Are you suggesting that you need something more for tensors that you don't need for say just matrices?
Right, ##A'=RAR^{-1}##. You can define a type (1,1) tensor by saying that that we associate each rotation matrix with a coordinate system, and that we for each rotation matrix R, associate the matrix ##RAR^{-1}## with the coordinate system associated with R. But you can also define a type (2,0) tensor by instead associating the matrix ##RAR## with the coordinate system associated with R.TGlad said:Matrices can be transformed between coordinate systems.