Andrius Kulikauskas

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Introduction E9F5FC

Questions FFFFC0


See: Math

Understand tensors and tensor products.


Read the chapter in Modern Geometry.

Understand tensor in the following way:

  1. What is a coordinate system?
  2. What is an invariant under coordinate transformation? (What is a tensor?)
  3. What are the arrays that express such an invariant, a tensor?
  4. What are the rules for reexpressing the tensor upon transforming the coordinates?

Ways of figuring things out

  • Understand the six interpretations of multiplication in terms of tensors (covariant, contravariant).

Understanding tensors

  • In what sense are tensors that which is invariant under change of frame?
  • In what sense are there two kinds of such invariance: covariance and contravariance?
  • Understand covariant derivative.
  • Study Riemannian geometry, Gaussian curvature, metric tensor.
  • Study Ricci tensor, general relativity.

标量, 向量, 矩阵


A tensor is:

  • that which is invariant under transformations of coordinates
  • an object best defined top-down. As does Daniel Chan, you first define what it means to have a space of such objects, and how to work with such space, products of such spaces, products of morphisms, and only then do you define the object accordingly.


A tensor product is:

  • the most general bilinear map that we can define on {$U\times V$}
  • relates two coordinate systems by matching the origins (by transfer of zeros) and allowing transfer of scalars (squeeze)


  • The image of such a map is not in general a vector space:

The image of {$*:K[x]\times K[y]\rightarrow K[x,y]$} contains {$x$} and {$y$} but not {$x+y$}.


Notes from Thomas Gajdosik

How do objects behave under transformations? and what are the quantities of the objects?

Tensor is describing the object.

A dipole is a vector and if you know how the vector behaves under rotation, then you know how the dipole behaves.

The object is represented as a tensor.

It behaves for us in a coordinate system, so it behaves as a multilinear array. Because we assume our transformations are multilinear.

Coordinate system is needed for the behavior and that is coming from the observer.

Invariants (when there are 2 indices): trace, symmetric part, antisymmetric part.

What are the invariant ways of formulating the tensor in multilinear that does not depend on the coordinate system.

My murky try at understanding tensors

A tensor is an algebraic object that is frame invariant, that is, invariant under (multi)(linear) changes in the reference frame.

Such invariance comes in two basic forms: covariant and contravariant.

Invariants of tensors are given by the characteristic polynomial (for rank two tensors) and related to symmetric functions.

Other notes

ML Baker video lecture about Tensors and tensor products

Lines (building up) are easily described by vectors - by construction, generation, spanning. One vector describes a line. We need more vectors to describe a plane, etc. Hyperplanes (tearing down) are easily described in terms of equations, by restriction, by conditions. One equation describes a hyperplane. We need more equations to describe a line, etc.

Constructing the most informative illustration of tensors.

Use 2x2 change in coordinates.

Use coordinate system for equilateral triangles and also a coordinate system for squares.


  • What is the manifold?


  • Bottom-up and top-down.
  • Tangent vector space and cotangent vector space.

Definition of a tensor:

  • A tensor of type (p, q) is a map which maps each basis f of vector space V to a multidimensional array T[f] such that if fR is another basis, then T[fR] = ...R-1...R T[f].
  • W:VxVx...xVxV*xV*...xV* -> R is a multilinear map (where V* is the dual space of covectors of the space V of vectors).

Determinant is top-down to define what is "inside" and what is "outside". A shape like the Moebius band is no fun because you can't make that distinction, you can't "understand" it, it does not make a "marked opposite". It is an unmarked duality. But for understanding we want a primitive marked duality, an irreducible marked duality. This is possible through the six transformations of perspectives given by the Holy Spirit.

Negative correlations vs. positive correlations. Yes vs. No.

Understanding the Lagrangian. Consider Kinectic Energy as "bottom-up" approach and Potential Energy as "top-down" approach. Kinetic Energy is finite and Potential Energy is possibly infinite. DT=−D is (roughly) anti-self adjointness.

  • Nature maximizes the explicit with regard to the infinite (thus kinectic energy with regard to potential energy). This minimization is related to the avoidance of the collapse of the wave function if at all possible. Nature prefers the complex (unmarked opposites) over the reals (marked opposites). Nature minimizes the marked opposites.

R is super rich but can't handle itself root wise, algebraically. But just a small "shift" is required to add unmarked opposites and have C. Unmarked opposites are "implications" rather than "explications".

Cramer's rule for inverses involves replacing a column in the matrix with the column with the constants. Replacing a column implies a "top down" orthogonal system. Also, the determinant is an anti-symmetric top-down system which distinguishes inside and outside. Whereas the symmetric case does not distinguish inside and outside and leaves them as unmarked opposites. In order to have marked opposites, we need to have a system of anti-symmetry.

Šešeriopai suvoktą dauginimąsi (multiplication) suvokti, išsakyti tensoriais.

Thank you very much for your discussion of dimensions. I'm grappling with this very much. But basically I think that you are describing the distinction between the "top-down" view of a space (in which we start with the whole space and break it down) and the "bottom-up" view of a space (in which we start with an empty space and build it up). That's what at's the heart of tensors. They combine the two points of view. They break up an n dimensional space into p bottom-up (contravariant = vector) and q bottom-up (covariant = covector = hyperplane = reflection) points of view.

I appreciate your thinking on this difference.

I've realized that I need to understand "tensors". They are quite central. They are a generalization of matrices to multiple dimensions. But truly the real point of tensors is that they break space into two different points of view, "top down" and "bottom up", or in other words, covariant and contravariant.

Imagine a tetrahedron (in 3 dimensions) with a natural coordinate space on it (given by three edges). That coordinate space, though, has angles that are not 90 degrees. Now consider removing one of these "bottom up" coordinate basis vectors and replacing it with a "top down" vector in the following way: choose a vector that would be perpendicular to the remaining basis vectors. This gives you an entirely different but "dual" basis. I suppose this comes up with the Platonic solids where the "dual" of the cube is the octahedron and vice versa.

Well, so tensors describe multidimensional space in terms of two sets of basis vectors. Rather than vectors it turns out it is more natural to think of them as maps (linear functionals) from the vector space to the real numbers. The tensor then "eats" say p many vectors and "yields" say q many vectors. (For a matrix it would be 1 vector in and 1 vector out, making it a linear transformation.)

These tensors are very important because, for example, if you want to do integration on a complicated manifold (multidimensional surface) you need that manifold to be oriented, which is to say, have an "inside" and an "outside", or a "left direction" and a "right direction", so that the integration would give opposite sign in either case. So that means you need a wedge product that is antisymmetric, that switches sign whenever you flip your area/volume/etc over, that is, whenver you swap coordinates.

Well, all of this to say that I'm realizing that tensors are key to "geometry" because they establish the geometric space between the top down vectors and the bottom up vectors. So I'm thinking that "geometry" is the way of embedding a lower dimensional space into a higher dimensional space.

Another insight that's helpful is that any matrix can be decomposed into two matrices, just like "polar decomposition", with length and angle. One matrix is "unitary" and it handles the rotations/reflections but keeps the lengths the same. The other matrix stretches the lengths shorter and longer as needed.

Lie groups are typically matrix groups which are also manifolds, so that the actions can be composed in a continuous way, even an infinitesimal way. You can rotate a circle or sphere just the slightest bit. It reminds me of the proof of the Pythagorean theorem given by the truth that "four times a right triangle is the difference of two squares". If the right triangles are long and thin, then one square is just the slightest rotation of the other. So Lie groups are the study of this sort of thing and Lie algebras are their infinitesimal rotations/changes.

So circles and spheres and etc. are very central in this subject.

Another idea that came up is that in the real numbers the dimensions (of space) are all independent. But what the complexes can be thought of as doing is "coupling" two independent dimensions with a coupling "i". And that i transforms by 90 degrees so that it swaps the y variable with the x variable and keeps track of that.


  • Tensors stay free of a coordinate system and work with all of them.
  • Tensor: function in all possible coordinate spaces such that it (its values) obey certain transformation rules.
  • There is a duality between a tensor and its expression under a particular basis. They are interchangeable.
  • Tensors are invariant under linear transformations but their components do change.
  • So a tensor is a bringing together of components, which can be either covariant or contravariant. Is this stepping out and stepping in? Is a tensor a division of everything and each component a perspective?

A tensor algebra is important for generating quotient algebras:

  • exterior algebra is the quotient by {$v\bigotimes v$} or alternatively {$v_1\bigotimes v_2 + v_2\bigotimes v_1$} so that {$v\bigotimes v = 0$}.
  • symmetric algebra is the quotient by {$v_1\bigotimes v_2 - v_2\bigotimes v_1$} so that {$v_1\bigotimes v_2 = v_2\bigotimes v_1$}.
  • Clifford algebra is the quotient by {$v\bigotimes v - Q(v)1$} where {$Q(v)$} is a quadratic form so that {$v\bigotimes v=Q(v)1$}.
  • Weyl algebra is the quotient by {$v_1\bigotimes v_2 - v_2\bigotimes v_1 -\omega (v_1,v_2)$} where {$\omega (v_1,v_2)$} is a symplectic form so that {$v_1\bigotimes v_2 - v_2\bigotimes v_1 = \omega (v_1,v_2)$}. The Weyl algebra is also referred to as the symplectic Clifford algebra. Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms.
  • Universal enveloping algebra` is the quotient by {$v_1\bigotimes v_2 - v_2\bigotimes v_1 - [v_1,v_2]$} so that {$v_1\bigotimes v_2 - v_2\bigotimes v_1 = [v_1,v_2]$}.

Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.


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This page was last changed on February 23, 2022, at 03:01 PM