Appel W. Mathematics for Physics and Physicists

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Tensors in afﬁne spac e 437

Then the family (αα

αα

)

is a basis of E

∗

, called the dual basis of (e

)

µ∈I

The linear forms (αα

αα

)

are also called coordinate forms, a nd often denoted

αα

= dx

; to each vector u = u

, the linear form dx

associates its coordinate u

: u = u

7−→ u

Those two deﬁnitions are of course consistent.

Proof of Theorems 16.9 and 16.10. Let ωω

ωω ∈ E

∗

be a linear form. For all µ ∈ [[1, n]],

denote ω

= ωω

ωω(e

). Then we have

ωω

ωω(u) = ωω

ωω(u

) = u

ωω

ωω(e

) = ω

αα

(u),

for all u ∈ E , showing that ωω

ωω = ω

αα

. Hence t h e family (αα

αα

)

is generating. It is

also free: if λ

αα

= 0, applying this form to the basis vector e

yields λ

= 0. So it is

a basis of E

∗

which consequently has the same dimension as E .

Since any linear form can be expressed uniquely in terms of the vectors of

the dual basis (αα

αα

)

, we make the following deﬁnition:

DEFINITION 16.11 (Covariant coordinates of a linear form) Let ωω

ωω be a lin-

ear form, that is, an element in E

∗

. The covariant coordinates of the linear

form ωω

ωω in the basis (αα

αα

)

are the coefﬁcients ω

appearing in its expression

in this basis:

ωω

ωω = ω

αα

(with the Einstein convention a pplied).

Hence the covariant coordinates of a linear form, just like the contravariant

coordinates of a vector, are its usual linear algebra coordinates.

They are computed using the following result:

THEOREM 16.12 (Expression of vectors and linear f orms) Let (e

)

be a basis

of E and let (αα

αα

)

be the dual basis of (e

)

. Then, for any vector u ∈ E and any

linear form ωω

ωω ∈ E

∗

, we have the decompositions

ωω

ωω = 〈ωω

ωω, e

〉αα

αα

= ω

αα

with ω

= 〈ωω

ωω, e

〉

u = 〈αα

αα

, u〉e

= u

with u

= 〈αα

αα

, u〉.

Proof. Indeed, we have

〈αα

αα

, u〉 = 〈αα

αα

, u

〉 = u

〈αα

αα

, e

〉 = u

= u

Similarly

〈ωω

ωω, e

〉 = 〈ω

αα

, e

〉 = ω

= ω

If the basis of E is changed, it is clear that the dual basis will also change.

Therefore, let e

′

= L

, wh ere L is an invertible matrix. We have then:

438 Tensors

PROPOSITION 16.13 If the basis of E is changed according to the matr ix L, the

dual basis is changed according to the inverse matrix Λ = L

−1

, that is, if we denote

(αα

αα

′µ

) the dual basis of (e

′

)

= L ·(e

)

, we have

′

= L

and αα

αα

′µ

= Λ

αα

for all µ.

Proof. We have indeed

αα

, e

′

αα

, L

= Λ

= δ

which proves that the family (Λ

αα

)

is the dual basis of (e

′

)

It is eas y now to ﬁnd the change of coordinates formula for a linear form

ωω

ωω = ω

αα

, using the same technique as for vectors. We leave the two-line

computation as an exercise to the reader, and state the result:

PROPOSITION 16.14 The covariant coordinates of a linear form are changed in the

same manner as the basis vectors, namely,

′

= L

Remark 16.15 The fact that the coordinates of a linear form are transformed in the same manner

as the vectors of the basis is the reason they are called covariant. Conversely, the coordinates of

a vector are changed in the inverse manner compared to the basis vectors (i.e., with the inverse

matrix Λ) and this explains why the coordinates are called contravariant coordinates.

Remark 16.16 Some quantities that may look like vectors actually transform like li near

forms. Thus, for a differentiable function f : E → R, the quantity ∇f transforms

as a linear form. The notation for th e gradient is in fact a slightly awkward. It is

mathematically much more e fﬁcient to consider rath er the linear form, depending on

the point x, given by

d f

∂ f

∂ x

(x) dx

(this is an example of a differential form, see the next chapter). This linear form associates the

number d f

(h), which is equal to the scalar product ∇f · h, to any vector h ∈ E . Hence,

to deﬁne the vector ∇f from the linear form d f |

, we must introduce this notion of scalar

product (what is call ed a m etric). We emphasize this again: because a metric needs to be

introduced to deﬁne it, the gradient is not a vector.

16.1.d Linear maps

Let Φ : E → E be a linear map. We denote (ϕ

)

µ,ν

its matrix in the bas is B.

Note that the row index is indicated by a super script, and the column index is indicated

by a subscript. The coefﬁcients are given by

def



αα

, Φ(e

)



Tensor product of vector spaces: tensors 439

and for any vector x, we have Φ(x) = ϕ

αα

(x) e

. Hence, under a change of

basis, t he coordinates of of Φ (i.e., its matrix coefﬁ cients) are transformed as

follows:

′µ

αα

′µ

, Φ(e

′

)

αα

, Φ(L

)

= Λ

PROPOSITION 16.17 (Change of coordinates) The coefﬁcients of the matrix of

a linear map are transformed, with respect to the superscript, as the contravariant

coordinates of a vector, and with respect to the subscript, as the covariant coordinates of

a linear form.

Thus, if a map is representated by the matrix ϕ in the basis B and by ϕ

′

in the

basis B

′

, we have

′

= Λ

This formula should be compared with the formula (c.2), on page 596.

 Exercise 16.1 Show that th e formulas

Φ(x) = ϕ

′µ

αα

′ν

(x) e

′

and Φ(x) = ϕ

αα

(x) e

are consistent. (Solution page 462).

16.1.e Lorentz transformations

As an illustration, we give here the matrices L and Λ corresponding to the

Lorentz “boost”: the frame of (e

′

) has, relative to the frame of (e

), a velocity

v = v e

. Denoting β = v/c and γ = (1 −β

)

−1/2

, we have

Λ = (Λ

) =







γ −βγ 0 0

−βγ γ 0 0

0 0 1 0

0 0 0 1







L = (L

) =







γ βγ 0 0

βγ γ 0 0

0 0 1 0

0 0 0 1







16.2

Tensor product of vector spaces: tensors

16.2.a Existence of the tensor product of two vector spaces

Note: Th is section may be s kipped on a ﬁrst reading; it contains the statement

and proof of the existence theorem for the tensor space of two vector spaces.

THEOREM 16.18 Let E and F be two ﬁnite-dimensional vector spaces. There exists

a vector space E ⊗ F such that, for any vector space G, the space of linear maps from

E ⊗ F to G is isomorphic to the space of bilinear maps from E × F to G, that is,

L (E ⊗ F , G) ≃ Bil(E × F , G).

440 Tensors

More precisely, there exists a bilinear map

ϕ : E × F −→ E ⊗ F

such that, for any vector space G and any bilinear map f from E × F to G, there

exists a unique linear map f

∗

from E ⊗ F into G such that f = f

∗

◦ϕ, which is

summarized by the following diagram:

E × F

E ⊗ F

∗

The space E ⊗ F is called the tensor product of E and F . It is only unique up to

(unique) isomorphism.

The bilinear map ϕ from E × F to E ⊗ F is denoted “⊗” and also ca lled the

tensor product. It is such that if (e

)

and (f

)

are basis of E and F , respectively,

the family (e

⊗f

)

i, j

is a basis of E ⊗ F .

All this means that the tensor product “linearizes what was bilinear,” and also

that “to any bilinear map on E × F , one can associate a unique linear map on the

space E ⊗ F .”

Proof. See Appendix D, page 604.

It is important to remember that the tensor product of two vector spaces

is not deﬁned uniquely, but always up to isomorphism. So there isn’t a single

tensor product, but rather inﬁnitely many, which are all isomorphic.

 Exercise 16.2 Let n, p ∈ N. For A = (a

i j

) ∈ M

(C), denote by A ⊗ B the

matrix of size np whic h is deﬁned by blocks as follows:

A ⊗ B =







B ··· a







Show th at the map (A, B) 7→ A ⊗ B gives an isomorphism

(np)

(C).

In particular, deduce that (E

i j

⊗ E

)

i, j,k,l

is a basis of M

(C). Show t hat there e xist

matrices in M

(np)

(C).

(Solution page 462).

THEOREM 16.19 If E and F are ﬁnite-dimensional vector sp aces, we have

dim E ⊗ F = dim E ×dim F .

Tensor product of vector spaces: tensors 441

PROPOSITION 16.20 In the particular case G = K, Theorem 16.18 gives

L (E ⊗ F , K) ≃ Bil(E × F , K).

For E and F ﬁnite-dimensional vector spa ces, we have

∗

⊗ F

∗

≃ Bil(E × F , K),

that is, the tensor product E

∗

⊗F

∗

is isomorphic to the space of bilinear forms on E ×F

(see Section 16.2.b).

16.2.b Tensor product of line ar forms:

tensors of type





The abstract construction in the previous section is made more concrete when

considering the tensor product of E

∗

and F

∗

, where E and F are two ﬁnite-

dimensional vector spaces.

DEFINITION 16.21 Let ωω

ωω ∈ E

∗

and σσ

σσ ∈ F

∗

be two linear forms on E

and F , respectively. The tensor product of ωω

ωω and σσ

σσ, denoted ωω

ωω ⊗σσ

σσ, is the

bilinear form on E ×F deﬁned by

ωω

ωω ⊗σσ

σσ : E ×F −→ K,

(u, v) 7−→ωω

ωω(u) σσ

σσ(v).

If (αα

αα

)

and (ββ

ββ

)

are bases of E

∗

and F

∗

, respectively, the family

(αα

αα

⊗ββ

ββ

)

µν

is a basis of the vector space of bilinear forms on E × F. In other words,

any element of Bil(E ×F , K) is a linear combination of the tensor products

αα

⊗ββ

ββ

. The space of bilinear forms on E ×F is a realization of the tensor

product E

∗

⊗F

∗

deﬁned in the previous section. This is denoted

Bil(E ×F , K) ≃ E

∗

⊗F

∗

and in general, with a slight abuse of notation, we identify t hose two s paces,

meaning that we do not hesitate to w rite an equal sign between the two.

Remark 16.22 A lthough it is perfectly possible to deal with the general c ase where E

are F are distinct spaces, we will henceforth assume t h at E = F , wh ich is the most

common sit u ation. Generalizing the formulas below causes no problem.

DEFINITION 16.23 A tensor of type









-tensor is any element in E

∗

⊗

∗

, in other words any bilinear form on E ×E .

442 Tensors

Coordinates of a





-tensor

THEOREM 16.24 Let T ∈ E

∗

⊗E

∗

be a





-tensor. It can be exp ressed in terms of

the basis (αα

αα

⊗αα

αα

)

µν

of E

∗

⊗E

∗

T = T

µν

αα

⊗αα

αα

with T

µν

def

= T(e

, e

Change of coordinates of a





-tensor

The coordinates of a





-tensor are transformed according to the following

theorem during a cha nge of basis:

THEOREM 16.25 Let T ∈ E

∗

⊗E

∗

be a tensor given by T = T

µν

αα

⊗αα

αα

. During

a basis change, the coordinates T

µν

are transformed according to the rule

′

µν

= L

ρσ

In other words, each covariant index of T is transformed using the matrix L.

Another viewpoint for a





-tensor

Since a





-tensor is (via the isomorphism in T heorem 16.20) nothing but a

bilinear form on E ×E , it is a “machine that takes two vectors in E and gives

back a number”:

T(vector, vector) = number.

Let’s continue with this mechanical analogy. I f we only put a single vector

in t he machine, what happens? Th en the machine expects another vector

(which we did not give) in order to produce a number. But a machine that

expects a vector to output a number is simply a linear form. Thus, once our

tensor is “fed” with one vector, it has been transformed into a linear form.

Precisely, if u ∈ E , the map

T(u, ·) : v 7−→ T(u, v)

is indeed a linear form on E .

This means that the tensor T ∈ E

∗

⊗ E

∗

may also be s een as a mach ine

that turns a vector into a linear form:

T(·, ·) : E −→ E

∗

u 7−→ T(u, ·)

Tensor product of vector spaces: tensors 443

which does indeed take a vector (denoted u) as input and “turns” it to the

linear form T(u, ·).

In mathematical terminology, showing th at a bilinear form on E × E





-tensor), is “the same thing as” a linear map from E (the space of

vectors) to E

∗

(the space of linear forms) is denoted

Bil(E ×E , R) ≃ E

∗

⊗E

∗

≃ L (E , E

∗

This property is clearly visible in the notation used for the coordinates of the

objects involved: if we write T = T

µν

αα

⊗αα

αα

, then we have

T(u, ·) = T

µν

αα

⊗αα

αα

(u, ·)

= T

µν

〈αα

αα

, u〉αα

αα

(= T

µν

〈αα

αα

, u〉〈αα

αα

, ·〉)

= T

µν

αα

which is a linear combination of the linear forms αα

αα

. The reader can check

carefully that thos e notations precisely describ e the isomorphisms given above,

and that all this works with no effort whatsoever.

A particularly important instance of bilinear forms is the metric, and it

deserves its own section (see Section 16.3).

16.2.c Tensor product of vectors:

tensors of type





DEFINITION 16.26 Let u ∈ E and v ∈ E be two vectors. The tensor product

of u and v, denoted u ⊗v, is the bilinear form on E

∗

×E

∗

deﬁned by

u ⊗v : E

∗

×E

∗

−→ K,

(ωω

ωω,σσ

σσ) 7−→ωω

ωω(u) σσ

σσ(v).

Any element in Bil(E

∗

×E

∗

, K) is a linear combination of the family (e

⊗

)

µν

We denote E ⊗E = Bil(E

∗

×E

∗

, K), and any element of E ⊗E is called

a tensor of type





or a





-tensor.

Coordinates of a





-tensor

The following theorem describes th e coordinates of a





-tensor:

THEOREM 16.27 Let T ∈ E ⊗E be a tensor. Then T may be expressed in terms of

the basis (e

⊗e

)

µν

of E ⊗E as

T = T

µν

⊗e

with T

µν

def

= T(αα

αα

,αα

αα

444 Tensors

Change of coordinates of a





-tensor

THEOREM 16.28 Let T ∈ E ⊗E be a tensor given by T = T

µν

⊗e

. During

a basis change, the coordinates T

µν

are transformed according to the rule:

′µν

= Λ

ρσ

In other words, each contravariant index of T is transformed using the ma-

trix Λ.

16.2.d Tensor product of a vector and a linear form:

linear maps or





-tensors

A special case of tensor product is the ca se of E ⊗ E

∗

, since th e previous

results allow us to write down is omorphisms

E ⊗E

∗

≃ Bil(E

∗

×E , K) ≃ L (E

∗

, E

∗

) ≃ L (E , E ).

Therefore, it should be possible to see the tensor product of a vector and a

linear form as a linear map from E into itself!

DEFINITION 16.29 Let u ∈ E and ωω

ωω ∈ E

∗

. Th e tensor product of the

vector u and the form ωω

ωω is the bilinear form on E

∗

×E deﬁned by

u ⊗ωω

ωω : E

∗

×E −→ K,

(σσ

σσ, v) 7−→ 〈σσ

σσ, u〉〈ωω

ωω, v〉.

Any bilinear form on E

∗

×E may be expressed as linear combination of the

family (e

⊗αα

αα

)

µν

. We write E ⊗E

∗

= Bil(E

∗

× E , K) and a ny element of

E ⊗E

∗

is called a tensor of type









-tensor.

Now we can spell out concretely the sta tement above, namely that:

E ⊗E

∗

≃ L (E , E ),

which gives an identiﬁcation of a





-tensor and an endomorphism of E .

Let T = T

⊗αα

αα

be such a tensor; then we can construct a n associated

linear map from E to E by

v 7−→ T

αα

(v) e

Conversely, a linear map may be seen as a





-tensor:

PROPOSITION 16.30 Let Φ : E → E be a linear map. Denote

def



αα

, Φ(e

)



Tensor product of vector spaces: tensors 445

A student of Jacobi and then of Dirichlet in Berlin, Leop o l d Kro-

necker (1823—1891) worked in ﬁnance for ten years starting when

he was 21. Thus enric h ed, he retired from business and dedicated

himself to math ematics. His interests ranged from Galois theory

(e.g., giving a simple proof that the general equation of degree at

least 5 cannot be solved with radicals), to elliptic functions, to

polynomial algebra. “God created integers, all the rest is t h e work

of Man” is his most famous quote. His constructive and ﬁnitist

viewpoints, opposed to those of Cantor, for instance, made him

pass for a reactionary. Ironically, with the advent of computers

and algorithmic questions, those ideas can now be seen as among

the most modern and lively.

its matrix coefﬁcients in the basis B = (e

)

. If (αα

αα

)

is the dual basis to B, then the

map Φ can also be seen as a bilinear for m on E

∗

×E given by

Φ = ϕ

⊗αα

αα

Proof. Indeed, we can show th at ϕ

⊗αα

αα

is equal to Φ by checking that it agrees

with Φ w h en acting on a basis vector e

. Since

⊗αα

αα

) =



αα

, Φ(e

)



⊗αα

αα

) =



αα

, Φ(e

)



〈αα

αα

, e

〉



αα

, Φ(e

)





αα

, Φ(e

)



= Φ(e

this is the case, and by li nearity it follows that Φ = ϕ

⊗αα

αα

DEFINITION 16.31 A tensor of type









-tensor is any element of

E ⊗E

∗

or, equivalently, any linear map from E to itself.

 Exercise 16.3 With the notation above, show that E ⊗E

∗

≃ L (E

∗

, E

∗

Notice, morevoer, that the coordinates of the identity map are the same in

any basis:

THEOREM 16.32 The coordinates of the identity map in any basis (e

)

are given

by the Kronecker symbol

def

1 if µ = ν,

0 if µ 6= ν.

Thus it is possible to write Id

= δ

and Id = e

⊗αα

αα

= e

⊗dx

, independently

of the chosen basis.

Remark 16.33 The formula “Id = e

⊗αα

αα

= e

⊗dx

” should be compared with the closure

relation given in Remark 9.26, page 260, namely Id =

〉〈e

|. The tensor product is

implicit there, and t h e basis (〈e

|) is the dual basis of (|e

〉) if t he latter is o rt h o normal.

446 Tensors

Change of coordinates of a





-tensor

The coordinates of a





-tensor are neither more nor less than the coefﬁ-

cients of the matrix of the associated linear map. Therefore, Proposition 16.17

describes the law that governs the change of coordinates of a





-tensor.

THEOREM 16.34 Let T ∈ E ⊗E

∗

be a tensor given by T = T

⊗αα

αα

. During

a basis change, the coordinates T

are transformed according to the rule

′

= Λ

Remark 16.35 In other words, each contravariant index of T is transformed using the matrix Λ

and each covariant index is transformed using the matrix L. Compare this formula wit h the

change of basis formula (equation (c.2), page 596) in li near algebra.

16.2.e Tensors of type





Of course, it is possible to continue the tensor process an arbitrary number

of times, either with E as a factor or with E

∗

. One has to be careful with

the order of the factors, since the tensor product is not commutative, strictly

speaking. However, there are canonical isomorph isms E ⊗E

∗

≃ E

∗

⊗E (i.e.,

isomorphisms which a re independent of the choice of a bas is) w hich justify

some simpliﬁcation, except that one must be quite careful when performing

“index gymnastics,” as described in the next section. It sufﬁces therefore,

when dealing with higher-order tensors, to consider those of t he type

p times

z }| {

E ⊗E ⊗···⊗E ⊗

q times

z }| {

∗

⊗E

∗

⊗···⊗E

∗

where the order of the factors is irrelevant.

DEFINITION 16.36 We denote E

⊗p

def

= E ⊗E ⊗···⊗E , where the tensor

product is iterated p times, and similarly E

∗⊗q

= E

∗

⊗. . . ⊗E

∗

(q times).

DEFINITION 16.37 A tensor of type









-tensor, is an element of

the space E

⊗p

⊗E

∗⊗q

; in other words, it is a multilinear form on the space

∗

×···×E

∗

×E ×···×E .

Coordinates of a





-tensor

THEOREM 16.38 Let T be a





-tensor. Then T may be expressed in terms of the

basis



⊗···⊗e

⊗αα

αα

⊗···⊗αα

αα



···µ

···ν

of the space E

⊗p

⊗E

∗⊗q

T = T

···µ

···ν

⊗···⊗e

⊗αα

αα

⊗···⊗αα

αα