About numbers, episode 5 : Tensors

Niveau de difficulté    

Update (12/09/2016) : several sections have been rewritten !

The (foolish) purpose of this post is to tackle the concept of tensor, while trying to keep it accessible to the widest audience possible. For the layman (or specialist from another field of study) who just wants to know what this is about without the rigidity and complexity of reading an indecipherable math handbook, for the student who already had a course on tensor calculus but can’t figure out some of the “obvious” stuff the teacher didn’t linger on, and finally for any math teacher or expert looking for a fresh eye on the subject (meaning not the handbook type) maybe to help describe it and teach it to his own audience.

It doesn’t mean there won’t be any formalism though, it will just be stripped down to the bare minimum while trying not to take too big of a shortcut when needed.

So, you will find a lot of visual representations along this post, that I hope will assist and guide you to understand each step, but keep in mind that visual aids are just that, they should not be the basis for understanding, only afterthoughts and guides.

Note : some paragraphs \color{red}{\Big[} in red square brackets\color{red}{\Big]} have been added or edited. The sections containing these changes are indicated by a (*) in the table of content.

Buckle up, you’re in for a ride !

A tensor is based on two concepts :
\star\quad There are several ways of looking at the components of a vector. We are first going to define them before trying to explain their subtleties and what this really means.

\star\quad There is a “mirror world” of vector spaces, and it’s the back-and-forth between these that will bring most of the defining properties of tensors.

These two seemingly simple points will help define what is a tensor of order one and help generalize the concept. It is necessary in order to really understand what they are made of and why they are so useful, particularly in physics.

Covariant and contravariant components of a vector

To fix the position of a point in a vector space, one needs as many numbers (real numbers, but not necessarily) as there are dimensions in this space. These are a point’s coordinates.

(1,1) for example gives us the position of a point distant of 1 from the origin. We will here (and through the whole post) use a real Euclidean vector space of two dimensions. So we have a scalar product, and an orthonormal basis e=\{e^{}_1,e^{}_2\}, that is to say a reference frame made of two orthogonal vectors of norm 1.

A vector is defined by its unique linear combination of the basis vectors (the unicity being of course due to the fact that we chose a basis of reference…).

Here, v=v^1\times e^{}_1 + v^2\times e^{}_2

So v=1\times \begin{pmatrix}1\\0\end{pmatrix} + 1\times \begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}1\\1\end{pmatrix}.

The v^i are the contravariant components of the vector v in the basis e (those are not exponents, they are just superscript numbers or letters). The main thing to remember here is that the contravariant components of a vector are the “number of times we take each vector of the basis”. They are the “building blocks” of the vector.
Note that from here, we’ll always write down these vector components in a column matrix, and we’ll see that there is a good reason for that convention.

We can define other components by projecting the vector onto each axis (meaning onto each vector of the basis), which is exactly what the scalar product was born to do :

v \cdot e^{}_1=\begin{pmatrix}1\\1\end{pmatrix} \cdot \begin{pmatrix}1\\0\end{pmatrix} =1
v \cdot e^{}_2=\begin{pmatrix}1\\1\end{pmatrix} \cdot \begin{pmatrix}0\\1\end{pmatrix}=1

These are the covariant components of the vector v (noted via subscripts) :

\begin{array}{cc} v^{}_1 =1\\v^{}_2=1\end{array}

The covariant components are the result of a projection, so they are the images of two linear forms applied to the vector.
Even if that sentence does not make sense right now, it will soon, so bear with me.

This covariant / contravariant distinction doesn’t seem to mean much right now because they are the same for this vector v. We will see that it is because of the orthonormal basis and the fact that we are using the usual scalar product. As an exercice you can take a different orthonormal basis and show that the covariant and contravariant components are still the same.

We are now going to see that in fact, those components are otherwise different.

Case of an orthogonal basis not normalized

Let’s take a situation where we can see this difference. Suppose we need to change the scale of our vectors by taking an orthogonal basis that’s “smaller” (by smaller, we mean that its vectors have a smaller norm than the first basis, when we express every vector in the standard basis, and it is not of norm 1, it is not normalized) b=\{\begin{pmatrix}\frac12\\[1em]0\end{pmatrix},\begin{pmatrix}0\\[1em]\frac12\end{pmatrix}\}

We can easily show that the contravariant components of the vector v become \begin{pmatrix}v^1\\v^2\end{pmatrix}=\begin{pmatrix}2\\2\end{pmatrix}

because \begin{pmatrix}1\\[.5em]1\end{pmatrix}=2\times \begin{pmatrix}\frac12\\[.5em]0\end{pmatrix}+ 2\times \begin{pmatrix}0\\[.5em]\frac12\end{pmatrix}.

While the basis “shrunk”, the contravariant components “grew bigger” : that’s why they’re called contravariant.

v^i=2\quad\text{in the basis}\quad (b^{}_i)

Note that the length (meaning Euclidean norm) of the vector hasn’t changed, only its relation to the basis has changed.

Its covariant components are then \begin{pmatrix}v^{}_1 & v^{}_2\end{pmatrix}=\begin{pmatrix}\frac12 & \frac12\end{pmatrix}

because \begin{pmatrix}1\\[.5em]1\end{pmatrix}\cdot \begin{pmatrix}\frac12\\[.5em]0\end{pmatrix}=\begin{pmatrix}1\\[.5em]1\end{pmatrix}\cdot \begin{pmatrix}0\\[.5em] \frac12\end{pmatrix}=\frac12.

So here it’s the contrary : the covariant component “shrunk”, just like the basis.

But, since the vector hasn’t changed, the fact that its components have “shrunk” means that they must be expressed in a “bigger” basis, so a different one…

Why did we change the basis ?

This is because to obtain our covariant components, we applied a scalar product, so we went into the dual space… We will come back to that in a moment. For now, let’s just say that the covariant components are in fact the dual vector’s components, which we note v^\star expressed in a new basis (b^i) :

v^{}_i=\frac12\quad\text{in the basis}\quad (b^i)

What’s a dual space ? That’s a very good question, thanks for asking.

Linear forms and the dual space

Linear forms are the “second heart” of tensors. Now we’re stepping on the accelerator it’s going to become abstract. We’re not in Kansas anymore, Toto.

A linear form is a sort of linear function on a vector space.
We are still in the “nice” Euclidean space \mathbb{R}^2. Here, a linear form is a function that takes a vector and transforms it into a linear combination of its components (giving us a real number).
Let’s take a vector w of components \begin{pmatrix}x\\y\end{pmatrix} and apply the linear combination x+2y. We note that linear form w^\star\colon\begin{pmatrix}x\\y\end{pmatrix}\mapsto x+2y and we get for example w^\star(\begin{pmatrix}4\\5\end{pmatrix})=4+2\times 5=14.

Forme lineaire 1 2

\color{red}{\Big[} A linear form is a projection : all the points situated on a particular arrow are sent to the same real number. A linear form is then a “direction of projection” in a way.
Note that we could have represented the real number line in a different position, that one is just convenient for visualization.\color{red}{\Big]}

Since a linear form is defined by a (unique) linear combination, we can represent it with two numbers, just like a vector. Here we have w^\star with components \begin{pmatrix}w^{}_1 & w^{}_2 \end{pmatrix}=\begin{pmatrix}1 & 2\end{pmatrix} that we’ll write in a row matrix. Just like vectors, linear forms “live” in a vector space of their own (that we are gonna call “dual space” of \mathbb{R}^2), and their components will depend on a basis (which will be, you guessed it, the dual basis).

Up until now, everything is fine. We’ve defined a new world, the dual space, but it is very similar to our well-known vector space, so nothing to fret about. By the way, linear forms are also called covectors. Figures.

BUT (yeah, it couldn’t last, could it ?), everything is reversed. The building blocks of a linear form are called its covariant components :

And that’s not all, because vectors and linear forms are also themselves said covariant or contravariant (and it changes depending on which country you studied math in, of course).
What a mess !
Well, not to worry. We are going to set things straight.

Let’s recap (*)

\star\quad \color{red}{\Big[} A vector is defined by the linear combination of its components with the vectors of the basis, so for v with contravariant components \begin{pmatrix}1\\2\end{pmatrix}, we have v=1\times b^{}_1 + 2\times b^{}_2, so :

When we change the basis, the vector has to “adapt” its components to compensate for it. This is so v does not change, and it’s why its (contravariant) components get “bigger” when the basis gets “smaller”. The vector v has an intrinsic existence : its existence does not depend on the basis, contrary to its components. Another way of seeing this is to consider that a vector is only defined in the standard basis, sort of “its only natural basis” in a way.

\star\quad A linear form is defined by the product of its components by the components of a given vector, so for the linear form v^\star of covariant components \begin{pmatrix}1 & 2 \end{pmatrix}, we have v^\star(\begin{pmatrix}x\\y\end{pmatrix})=1\times x+2\times y.
But since the vector is expressed in a given basis, we only need to know how the linear form acts on the basis’ vectors, so for each i we have v^\star(b^{}_i)=\begin{pmatrix}1 & 2 \end{pmatrix}b^{}_i which we can sum up in :

Here when we change the basis, the linear form doesn’t have to compensate anything and their components vary in the same way as the basis.
Remark : We are here in a euclidean space where \cdot represents the usual scalar product. In a more general quadratic space (or even in a Euclidean space where the scalar product is different), we would have to use the corresponding symmetric bilinear form.When we use the scalar product to project a vector onto a basis vector to get its covariant component, we are in fact “switching” to the dual space : the projections of v onto the basis vectors are linear forms, and the results of these projections are the covariant components in the dual basis. These are by definition the components of the dual vector in the dual basis. That’s why we changed from the blue basis to the red one.\color{red}{\Big]}

Link between vector and linear form (*)

Let E=\mathbb{R}^2 be our Euclidian vector space, and E^\star its dual.

\color{red}{\Big[} Here, it is the scalar product that is going to link the elements of E and those of E^\star. More precisely, it induces an isomorphism :

This mapping gives us a one-to-one correspondance between E and E^\star. For each vector v\in E, there is a unique linear form v^\star\in E^\star. The two spaces are said to be isomorphic.
Not surprising that they look so much alike !

Important notes :

\star\quad This correspondance depends on the basis chosen to describe v !
Indeed, {v^\star(b^{}_1)=v^{}_1} and {v^\star(b^{}_2)=v^{}_2} are by definition the components of the dual vector in its dual basis, and are clearly dependant on the basis {b}.

\star\quad Since we are in that particular Euclidian space where v\cdot v'=xx'+yy' (the usual scalar product), it follows that v and v^\star have the same componants in the standard basis, they are “the same”, or “superposed” if we draw them in the same graph. This is a canonical correspondance between E and E^\star but it is by no means the only one.
If we had a different scalar product, like v\cdot v'=2xx'+3yy' for example, the two would then have different components in the canonical basis and then different covariant and contravariant components in general. This would be a different correspondance between E and E^\star.

\star\quad In the more general case of a quadratic space (E,\phi) where \phi is a non-degenerate bilinear symmetric form, not only would v and v^\star have different components, but every use of the scalar product \cdot would have to be replaced by \phi. It would then again be a completely different correspondance between E and E^\star.
The Minkowskian space of special relativity would be an example of that case. \color{red}{\Big]}

Links between contravariant and covariant components

Let’s get back to business. We have a vector v=v^i b^{}_i (definition of a vector as a linear combination of its basis)

Note that we are now using Einstein’s notation : v^i b^{}_i=\displaystyle\sum_{i=1}^2 v^i b^{}_i=v^1 b^{}_1+v^2 b^{}_2.
This notation only consists in removing the summation symbol, but beware : only opposite indices are summed over (one superscript and one subscript). Never sum aver identical indices if they are at the same level.

\color{red}{\Big[} In our example, v=\begin{pmatrix}1\\1\end{pmatrix}\quad \text{(in the standard basis),}
(v^i)=\begin{pmatrix}2 \\ 2\end{pmatrix}\quad \text{in the basis }\{b^{}_i\}=\{\begin{pmatrix}\frac12\\0\end{pmatrix},\begin{pmatrix}0\\\frac12\end{pmatrix}\}
So we indeed have v=v^i b^{}_i.\color{red}{\Big]}

By taking the scalar product by the b^{}_j of the vector, we get
v\cdot b^{}_j=(v^i b^{}_i)\cdot b^{}_j
v^{}_j=(b^{}_i\cdot b^{}_j)v^i
We indroduce a special notation for the scalar products of the basis vectors : g^{}_{ij}=b^{}_i\cdot b^{}_j
So
v^{}_j=g^{}_{ij}v^i

Just to be clear : g is a matrix, where the component at the intersection of line i and column j is b^{}_i\cdot b^{}_j.

\color{red}{\Big[} Back to our example, \begin{pmatrix}\frac12 & \frac12 \end{pmatrix}=g\begin{pmatrix}2 \\ 2 \end{pmatrix}

We have g^{}_{11}=b^{}_1\cdot b^{}_1=\begin{pmatrix}\frac12\\0\end{pmatrix}\cdot\begin{pmatrix}\frac12\\0\end{pmatrix}=\frac14 and g^{}_{22}=b^{}_2\cdot b^{}_2=\begin{pmatrix}0\\\frac12\end{pmatrix}\cdot\begin{pmatrix}0\\\frac12\end{pmatrix}=\frac14 and g^{}_{12}=g^{}_{21}=0

So g=\begin{pmatrix}\frac14 & 0 \\ 0 & \frac14 \end{pmatrix}.

We can also show that v^j=g^{ij}v^{}_i with g^{ij}=b^i\cdot b^j making the components of the inverse matrix g^{-1}.

Let’s calculate the components of the dual basis’ vectors which we’ll note b^\star=\{b^1,b^2\}.

We then have b^i b^{}_j=\delta^i_j so then b^1=\begin{pmatrix}2\\0\end{pmatrix} and b^2=\begin{pmatrix}0\\2\end{pmatrix}

And while we’re there g^{-1}=\begin{pmatrix}4 & 0 \\ 0 & 4 \end{pmatrix}.

We just verified that b^\star is in fact “bigger” than the original basis. Mystery solved.

Let’s remember that v^\star=v^{}_i b^i

v^\star=\begin{pmatrix}1 & 1\end{pmatrix}\quad \text{(in the standard dual basis),}
(v^{}_i)=\begin{pmatrix}\frac12 & \frac12\end{pmatrix}\quad \text{in the dual basis }\{b^i\}=\{\begin{pmatrix}2 & 0\end{pmatrix},\begin{pmatrix}0 & 2\end{pmatrix}\} \color{red}{]}

Note : this can be generalized. Had we “shrunk” only one vector of the basis in E, say b^{}_1, the dual basis vector b^2 would have “grown”, but b^1 would have stayed the same. Had we rotated one vector, it would have been the other one’s dual that would have reflected that rotation (in the same direction). There is a symetry, a sort of “mirror effect” between a basis and its dual, so that we always have b^{}_ib^j=\delta^j_i and that’s what generates this reversal.

Beware ! Here the dual space is defined via the scalar product, so a vector and its dual are always superposed (except for the basis). With another scalar product, or a pseudo scalar product like in the case of Minkowski’s space, the correspondance would be different and the vectors would not be superposed. The metric tensor and the scalar product’s associated bilinear form are often confused with one another, but it is very wrong… The two are usually different (though we are particularly fond of the case where they are the same, hence the confusion).
To sum-up, the dual space depends on the scalar product used (or quadratic form), but the one-to-one mapping between the two spaces (the metric tensor) depends not only on the scalar product but also on the chosen basis !

So we now have the means to go directly from one type of components to the other (that is to say from the space to its dual). Let’s see what this g is exactly.

First, note that in the case of an orthonormal basis (norms=1 and orthogonal), the scalar products of each couple of basis vectors would be zero if they’re different (i\neq j) and equal to 1 otherwise (when i=j). Here g is simply the identity matrix. We then have v^i=v^{}_i which means equality between the covariant and contravariant components.
Remark : the notation {v^i=v^{}_i} is an abuse, we do not sum indices here !

Now, it’s not going to be as easy in our “shrunken” basis. To understand what the matrix g is exactly, we will need to take a look at changes of basis. When we go from a basis e to a basis e', we can define a new basis matrix P so that e^{'}_i=Pe^{}_j
Say what ? Don’t we usually express the old basis in function of the new one ? Yes, but it’s confusing. Let’s do thing in their natural order instead of writing conveniance.

With that in mind, the contravariant components of a vector v will obey v^{'i}=P^{-1}v, they “vary contrary to the basis” because the matrix is inverted.

The covariant componants will then verify v^{'}_i={}^tPv, they “vary like the basis”.
Careful : here we are using the usual scalar product, so we should really write v^{'}_i={}^tPIv in order to not forget that (or v^{}_i={}^tPv^\star which is the exact same thing).

From here, we can say in a way that the g^{}_{ij} and g^{ij} matrices are the “new basis matrices” between the basis of our vector space and the dual basis of its dual space (it would be the identity matrix in the case of an orthonormal basis in a Euclidean space).

In a way, these matrices transform a vector into a linear form (and vice-versa) by lowering or raising their indices. They transform contravariant components into covariant components (and vice-versa).

If we use the matrix notation, we have \begin{pmatrix}v^{}_1 & v^{}_2 \end{pmatrix}=g^{}_{ij}\begin{pmatrix}v^1 \\ v^2 \end{pmatrix} and \begin{pmatrix}v^1 \\ v^2\end{pmatrix}=g^{ij}\begin{pmatrix}v^{}_1 & v^{}_2 \end{pmatrix}.

This g is a weird matrix (even when it’s the identity). Usually, a matrix represents a linear transformation (be it invertible or not) that applies to a vector (column matrix in our notation) and produces another vector. If we follow strictly the rules of matrix multiplication, we cannot apply a matrix to a linear form (row matrix) !

The reason is simple : g is a tensor. So it looks a lot like a usual linear transformation (and it partly is), but works a bit differently. The matrices of linear transformations are tensors with two indices, but one is covariant, the other contravariant. g is different : it is two times covariant in the form g^{}_{ij} and two times contravariant when in the form g^{ij}.
So it’s not so simple as to identify g and its matrix. We should rather say its components table, or its associated matrix that applies not to linear forms but to their associated vectors in the dual space (then it’s also an abuse to call the elements of the dual space linear forms, but let’s not dwell on that technicality that would complicate things unnecessarily, with my apologies to those who would take offense^^)…

When g is two times covariant it transforms a vector into a linear form, and vice-versa when it is two times contravariant. That’s a different beast !

g is called the metric tensor.

Ok now, it is time to simplify all that “components mess” thanks to the dual space :

Change of dual basis covariant and contravariant components

In conclusion, by passing to a “smaller basis”, the contravariant components of the vector v got “bigger”, while its covariant components (which are also the contravariant components of its associated linear form or dual vector) got “smaller”.

Bonus notes :

\star\quad \color{red}{\Big[} Because a vector’s indices are always subscript it makes sense to say that a vector is covariant while its components are not. On the other hand, the indices of a linear form (or dual vector or covector) are always superscript (contrary to its components), so it makes sense to say that a linear form is contravariant while its components are covariant.
But this has nothing to do with a change of basis, since both of them are defined in their respective standard basis. It is merely a convention. \color{red}{\Big]}

\star\quad Of course, the mathematicians from all over the world decided to confuse the students a little bit more by taking the converse convention in some countries, where the vectors are said contravariant because their components are, and the linear forms covariant for the same reason… That’s diabolical !

Let’s do it again in a more complicated case (just barely). It will also be more interesting and more visual too !

The case of a non orthogonal basis

This time, we’ll take a basis system f in which the vectors make a \frac{\pi}3 angle. For example we can take f=\{\begin{pmatrix}1\\[.5em]0\end{pmatrix},\begin{pmatrix}\frac12\\[.5em] \frac{\sqrt3}2\end{pmatrix}\}
Let’s use the same vector v of components \begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}1\\1\end{pmatrix} in the (orthonormal) standard basis.

Its contravariant components are v^i=\begin{pmatrix}\frac{3-\sqrt3}3\\[.5em] \frac{2\sqrt3}3\end{pmatrix}

because \begin{pmatrix}1\\1\end{pmatrix}=v^1\begin{pmatrix}1\\0\end{pmatrix}+v^2\begin{pmatrix}\frac12\\[.5em] \frac{\sqrt3}2\end{pmatrix}

Its covariant components are v^{}_i=\begin{pmatrix}1 & \frac{1+\sqrt3}2\end{pmatrix} via the scalar products v\cdot f^{}_i

Now, to represent them in a graphic, we need the dual basis’ vectors.
Using the usual scalar product, they are defined by the following property : f^i(f^{}_j)=\delta^i_j with \delta^i_j=1 if i=j and 0 otherwise. Visually, it means that the dual basis f^\star is orthogonal to the f basis (that is part of the reason why the components are linked the way they are, and the reason why we should never represent the covariant and contravariant components in a common basis, contrary to widely used erroneous representations).

That gives us the dual basis system f^\star=\{\begin{pmatrix}1 & -\frac{\sqrt3}3\end{pmatrix},\begin{pmatrix}0 & \frac{2\sqrt3}3\end{pmatrix}\}.

Here we can clearly see that the contravariant components v^i are expressed in the f^{}_j basis, while the covariant components v^{}_i are expressed in the dual basis f^j. Also, one reads the components by parallel projection since we express the vectors by linear combination of the basis vectors : remember the Chasles relation of vector addition ?

Of course, since the dual basis is orthogonal to the first base, each dashed line is perpendicular to the axis of the other base. So one needs to be careful when representing them in a common graph, and add both basis systems, even if it’s a little bit heavy on the eyes :

Here is a visual representation of the projection used to depict the covariant components :

The orthogonal projection in a space is always equivalent to the parallel projection in its dual space…

Correct representation of covariant and contravariant components

We then have a correct visual representation of these two types of components (in a Euclidean space may I remind you). It is rather intuitive that in a “squeezed” basis, we can read coordinates and components in two different ways : by orthogonal projection (covariant components) or by parallel projection (contravariant components). But beware ! There is a trap in the left-hand picture : the orthogonal projection is a parallel projection on the other base… The right way to do it is depicted on the right-hand side.

\color{red}{\Big[} Important reminder : What of a non-Euclidean space or on with a non usual scalar product ? Well, then v and v^\star would not be superposed anymore… So beware, in that representation we are here in a very special case ! In particular, that representation does not apply to a Minkowski space. So any such representation of the components of vectors in special relativity would be erroneous…\color{red}{\Big]}

Let’s take a look at matrices (*)

Fasten your seat-belt. No, really do, it’s going to get bumpy.

Let v be a vector in the standard basis, and A a linear map that sends this vector unto another vector w (still in the standard basis). We write P the change of basis matrix. The tensors of rank two map their respective components :

In the convenient case where A is invertible, we then have a linear map B=A^{-1} and then B^{ki} is the inverse tensor of A^{}_{ik} and B^{}_{\ell j} is the inverse of A^{j\ell} and again B^{i}_{\quad \ell} that of A^{\quad \ell}_i

Remark : The -1 exponent must be reserved only for matrices or linear maps and never be used with tensors, because the inverse of a bilinear form (associated with the matrix A in the standard basis) is not well defined, one must instead consider the bivector associated with the inverse matrix A^{-1} in the standard basis.
The inverse tensor of A^{}_{ik} only exists if A is invertible and it is then B^{ki}. The letter B is used to indicate that this tensor is associated with the matrix B=A^{-1} after a change of basis, and not with the matrix A.

As proof, A_{ik}={}^tPAP and B^{ki}=P^{-1}B{}^tP^{-1} so it follows that A_{ik}B^{ki}=I.
The notation B^{ki} for the inverse tensor is the only correct notation, since the redondancy of (A^{-1})^{ki} or (A^{ki})^{-1} would certainly confuse things. For practical use however, we note A^{ki} even if it technically is an abuse.

Here they are :

\star\quad A^{}_{ik} is a tensor two times covariant. It is a bilinear form that takes a vector and returns a linear form, or that transforms two vectors in a scalar (x,y) \longmapsto {}^txAy. In a change of basis, we have (A^{}_{ik})'={}^tP(A^{}_{ik})P. The matrices A^{}_{ik} and A^{'}_{ik} are called congruent (invariant : the rank !). In our example, we define A^{}_{ik}={}^tPAP and applied to the contravariant vector, we get A^{}_{ik}v^i=w^{}_k

\star\quad A^{\quad \ell}_{i} is a usual matrix, one time covariant and one time contravariant (in that order). It is a linear transformation, that takes one vector and gives back another vector. In a change of basis, we have (A^{\quad \ell}_{i})'=P^{-1}(A^{\quad \ell}_{i})P.
The matrices A^{\quad \ell}_{i} and (A^{\quad \ell}_{i})' are said similar (remark : if we take two different vector spaces, we would then have two different change of basis matrices and A^{\quad \ell}_{i} and (A^{\quad \ell}_{i})' would be called equivalent).
Similarity invariants : the rank, the determinant, the trace, the eigen values, the characteristic polynomial, the minimal polynomial, the Jordan form (a complete set of invariants), the Young tables…
In our example, we define A^{\quad \ell}_{i}=P^{-1}AP and applied to the contravariant vector, we get A^{\quad \ell}_{i}v^i=w^{\ell}

\star\quad \color{red}{\Big[}A^{j}_{\quad k} is also a usual matrix, one time contravariant and one time covariant (in that order). It is a linear transformation, that takes one linear form and gives back another linear form. In a change of basis, we have (A^{j}_{\quad k})'={}^tP(A^{j}_{\quad k}){}^tP^{-1}.
But… this is not typical of a linear application ! Some would say. Be not afraid, take the transpose…
The transposes of matrices A^{j}_{\quad k} and (A^{j}_{\quad k})' are similar.In our example, we define A^{j}_{\quad k}={}^tPA{}^tP^{-1} and applied to the covariant vector, we get A^{j}_{\quad k}v^{}_j=w^{}_k

Note : If A is symmetric, then A^{j}_{\quad k} is nothing but the transpose of the previous tensor A^{\quad \ell}_{i}\color{red}{\Big]}

\star\quad A^{j\ell} is a tensor two times contravariant. It is a transformation that takes a linear form and gives back a vector, or takes two linear forms and gives back a scalar. In a change of basis, we have (A^{j\ell})'=P^{-1}(A^{j\ell}){}^tP^{-1}. I suggest to call these matrices A^{j\ell} and A^{'j\ell} conbluar. No ? similuent then ?
In fact these matrices are congruent via {P^{-1}}
In our example, we define A^{j\ell}=P^{-1}A{}^tP^{-1} and applied to the covariant vector, we get A^{j\ell}v^{}_j=w^{\ell}

Here we can clearly see the bivalent role of A. It can be seen either as a linear map, a bilinear form or a bivector, because in the standard basis, they are all described by the same matrix (the dual standard basis is itself).
However, as soon as we change basis it takes one particular form depending on the type of change it is. A “acquires” its covariant/contravariant properties depending on the basis.
And for dessert, what links them all : the metric tensor of course since g^{}_{kj}g^{ik}=\delta^{i}_{j} gives us A^{}_{ij}=g^{}_{ki}A^{k}_{\quad j}=g^{}_{kj}g^{}_{li}A^{lk}.

All of the metric tensor’s dirty secrets

Let’s rotate our favorite vector \vec v=\begin{pmatrix}1\\1\end{pmatrix} by an angle of \dfrac{\pi}2. To do so, we just have to multiply it by the rotation matrix R=\begin{pmatrix}0&1\\-1&0\end{pmatrix} to get our new rotated vector w=Rv=\begin{pmatrix}1\\-1\end{pmatrix}.

The new basis will be f=\{f^{}_1,f^{}_2\}=\{\begin{pmatrix}1\\0\end{pmatrix},\begin{pmatrix}\frac12\\[.5em] \frac{\sqrt3}2\end{pmatrix}\}. We then have all we need to determine our change of basis matrix P so that f^{}_j=Pe^{}_j.

The contravariant components of v will be v^i=P^{-1}v and its covariant components v^{}_i={}^tPIv (The I is the identity matrix, here only to remind us that we used the usual scalar product). The dual basis components will then be f^j={}^tP^{-1}e^{}_j. (let’s not forget that e^{}_j=e^j since e is the standard basis)

Bonus :

We can easily check that g^{}_{ij}=(g^{ij})^{-1} and v^{}_i=g^{}_{ik}v^k.

\color{red}{\Big[} What use is the identity matrix here ? Well, it’s just a reminder that in a more general case of a quadratic space, one just has to replace I by the matrix of the symmetric bilinear form…\color{red}{\Big]}

Now, let’s see the components of w in the new basis without using the metric tensor (no real reason except to see what becomes of the rotation matrix) :

For the contravariant components, it’s easy, we changed basis, so we are going to do the same for the rotation matrix which will then become R'=P^{-1}RP and w^i=R'v^i.
This change of basis is due to the fact that the rotation matrix is a mixed tensor, one time covariant and one time contravariant. \color{red}{\Big[} Another notation would be R'=R^{\quad \ell}_{i} (careful of the indices’ order !).\color{red}{\Big]}

Now for the covariant components, it’s a little bit different. We would think that using the metric tensor would be enough to raise or lower one index of the tensor R', but that’s not the case…
In fact, the rotation matrix applied to v^{}_i will be R''=QRQ^{-1} with Q={}^tPI
We have to make a change of basis “in the other direction” and by transposing.
\color{red}{\Big[} Or again R''=R^{j}_{\quad k}.\color{red}{\Big]}

We have then a change of basis matrix P going from e to f, and {}^tP^{-1}I going from e to f^\star.
In both cases we use the matrix P, but to get to the dual space, we need to invert rows and columns and go in the opposite direction. In a way, the first space is generated by the matrix’s columns, and the dual space by the inverse matrix’s rows.

But let’s not stop just there. Let’s go a step further to understand what that means.

Some types of change of basis will give us the same metric tensor. For example, orthogonal matrices in a Euclidean space verify

so each time the change of basis matrix will be, say, a rotation for example, the metric tensor will be the identity. Of course, because a rotation does not change the orthonormality of the standard basis in a Euclidean space !

Now say we are in a Minkowski space. That is to say we replace the scalar product with a pseudo scalar product represented by I^{}_{3,1} the diagonal matrix which has three 1 and one -1 or vice-versa depending on the convention (now we are obviously in a four-dimensionnal space but it doesn’t change any formula). For the physicists, this form’s little name is \eta^{}_{\mu\nu}, called the Minkowski form.

Then, when the change of basis matrix is a general Lorentz transformation \Lambda\in O(3,1), we will get Minkowski’s tensor metric because 

This rotation would then become R'=\Lambda^{-1}R\Lambda to transform the contravariant components of vectors, and R''=QRQ^{-1} with Q={}^t\Lambda\eta^{}_{\mu\nu} to transform covariant components.
Note that I has here been replaced by \eta^{}_{\mu\nu}.

Conclusion : as long as we are making changes of basis that belong to the same class of transformations (i.e. generate the same metric tensor), we have unique formulas to effect these changes wihtout even knowing the basis vectors !

That is THE big advantage of tensors : no need to take into account each change of basis individually, we only have to consider a set or class of such changes thanks to the metric tensor !

Boom.

What is a tensor ?

Now we finaly have all the tools we need to answer this question.

Tensors are not just the generalization of the concepts of vectors, matrices and multidimensionnal tables of numbers. They are first and foremost a generalization of the concept of linear form (and multilinear forms). They combine the advantages of both objects : the contravariant components conserve linear combinations (just as vectors do), and the covariant components conserve their proportions relative to the basis (just as linear forms do).
Tensors are dual objects (even when they’re of order one), as in a person and his image in the mirror.

This duality gives tensors a relative independance from the basis system : we define a set of changes of basis that correspond to a unique metric tensor, and we will be able to easily effect these changes on tensors.

This brings joy to the physicists, because it allows them to describe the laws of nature so that they do not change for a different spectator, a change that is reduced to a change of basis linked to a particular metric tensor, and that is fondamental.

The concept of covariance in physics

Physicists have two goals :

\star\quad Defining quantities that do not change with a change of frame (inertial frames in special relativity for example – in this case they are said “Lorentz invariant”)

\star\quad Modeling the laws of physics in the form of relations (equations really) that do not change with a change of frame. These will be relations between tensors of the same type, or between invariant quantities. We say that these equations are “covariant”.

Note : A common misuse is to label physical quantities as covariant (or not covariant). This generates a lot of confusion… Because if this quantity is a scalar or a tensor in Minkowski space (and whatever type of tensor, contravariant or covariant), then it will necessarily be Lorentz covariant ! On the other hand, if it is not a tensor, or if it is a tensor from a different space (for example in three dimensions) then it would not be Lorentz covariant. It’s as simple as that and doesn’t need extra complications.

This invariance or covariance is relative to a particular group of transformations that depicts the types of change of basis “authorized” (a set chose precisely so they would all have the same metric tensor and be in that way equivalent) :

\star\quad In special relativity, the equations have to stay the same with any change of inertial frames (frames in a straight and uniform mouvement), so they have to be covariant under transformations of the Lorentz group. Same thing for quantum electrodynamics. For example, the relation \vec{p}=m^{}_0\vec{v} describing momentum as the product of rest mass by the speed \vec{v}=\begin{pmatrix}\frac{dx}{dt}\\[2ex] \frac{dy}{dt} \\[2ex] \frac{dz}{dt}\end{pmatrix} in 3 dimensions does not stay the same under Lorentz transformations (\vec{p} is not Lorentz covariant). Adding a time component is not enough to correct this (that’s why special relativity is so much more than just the union of space and time…), one also needs to play with the proper time \tau introduced in special relativity, and we then have a new definition of the momentum, called the four-momentum vector \mathbf{p}=m^{}_0\mathbf{u} where \mathbf{u}=\begin{pmatrix}c\frac{dt}{d\tau}\\[2ex]\frac{dx}{d\tau}\\[2ex] \frac{dy}{d\tau} \\[2ex] \frac{dz}{d\tau}\end{pmatrix} is the four-velocity vector. By the way, it gives us \mathbf{p}=\begin{pmatrix}E/c \\ \vec{p}\end{pmatrix} linking momentum and energy.
The relation \mathbf{p}=m^{}_0\mathbf{u} is then Lorentz covariant.
Note that even if the relation stays the same, the tensors components will usually change with a change of frame (but always in the same way, keeping the relation intact). A “real” invariant would be its pseudo-norm which will have the same value in every frame. For example with the four-velocity \mathbf{u}^{\mu}\mathbf{u}^{}_{\mu}=c^2, or for the four-momentum \mathbf{p}^{\mu}\mathbf{p}^{}_{\mu}=\dfrac{E^2}{c^2}-\vec p^2=m^{2}_0c^2 and these equations are not only Lorentz covariant, they are also completely invariant (their values do not vary).
By the way, at rest (\vec p=0) we obtain the famous equation E=m^{}_0c^2, and, that’s precisely the point, in every reference frame.

\star\quad In general relativity, the equations have to stay the same under any change of frame (and not just inertial ones). These changes are described this time by the group of diffeomorphisms of space-time, and the best way to get generally covariant relations is to use tensor fields (but it is quite a bit more complex to generalize the covariance of equations in general relativity, because the derivation does not transform a tensor in another tensor, so one has to define a covariant derivation…).

Zoology : examples of tensors

A tensor is characterized by its order and valence :
The order corresponds to the number of indices. For each index, it needs as many numbers as there are dimensions, so a total of n^{order} numbers. When indices are not used, the tensor is underlined as many times as its order, for example T^{j}_i=\underline{\underline{T}}.
The valence (h,k) gives us the number of contravariant indices h and the number of covariant indices k.
Note that in a change of basis, the valence gives us the number of times {k} we have to multiply by the change of basis matrix, and the number of times {h} we have to multiply by its inverse.

Tensors of order 0 : these are the scalars. Real numbers in a real space, or complex numbers or others depending on the field… They are “usual” numbers without indices, that do not depend on any basis system and then do not need any notion of covariance or contravariance.

Tensors of order 1 : these are the vectors (T^i with contravariant components) and covectors or linear forms (T^{}_i with covariant components).

Tensors of order 2 : these are the matrices T^i_j (linear transformations) or tensors of valence (1,1), tensors of valence (2,0) (bivectors, like the inverse metric tensor g^{ij}, the electromagnetic tensor, etc.), and finally the tensors of valence (0,2) (bilinear forms or bi-covectors, like the scalar product via its associated bilinear form, the trace, the metric tensor g^{}_{ij}, the Ricci tensor, symplectic forms, etc.).

We also find in this category numerous physical quantities : the magnetic field, magnetic flow, angular momentum, torque, angular velocity, angular acceleration, areal velocity, etc. Also called spinors, girators, axial vectors, pseudovectors… there is a vast terminology.
Note : the term “quadrivector” or “four-vector” used by physicists does not correspond to a tensor of order two, but only to a vector in four-dimensional space.

Tensors of order 3 : these are tensors with three indices, written T^{ijk}, T^{}_{ijk}, T^{i}_{jk}, T^{ij}_{k}.

Etc…

A word of philosophy

If you are surprised that this post is part of the “About numbers” series, then you probably make a clear distinction between the concept of number (natural, integer, rationnal, real, complex, etc.) and that of more abstract mathematical objects. I do not. I won’t discuss it in length, but let me say a few words.

First, every concept of number can be described as a vector in a particular space… Complex numbers are an example (more on that coming in a future post), but real numbers are another (in a trivial one-dimensionnal space), integers and rationnal numbers too (in a lattice), etc.

Also, vectors, matrices and tensors’ components are numbers, so they can be seen as multi-dimensionnal numbers in a way (even if it does not tell the whole story).

So the concept of number takes a very, very broader sense in mathematics than intuition usually tells us…
But, a table of number is not a number ! I hear you saying.

To that I’ll just answer : \begin{pmatrix}1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\end{pmatrix}, \begin{pmatrix}2 & 0 & 0 & 0 &0\\ 0 & 2 & 0 & 0 &0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 &2\end{pmatrix}, \begin{pmatrix}3 & 0 & 0 & 0 &0\\ 0 & 3 & 0 & 0 &0\\ 0 & 0 & 3 & 0 &0 \\ 0 & 0 & 0 & 3 & 0\\ 0 & 0 & 0 & 0 & 3\end{pmatrix}, etc…

 ;-)

Feedback greatly appreciated, thanks !

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