Universitext
Juan Ramon Ruiz-Tolosa
Enrique Castillo
From Vectors
to Tensors
Springer
Authors
Juan Ramon Ruiz-Tolosa
Enrique Castillo
Universidad Cantabria
Depto. Matematica Aplicada
Avenida de los Castros
39005 Santander, Spain
castie@unican,es
Library of Congress Control Number: 20041114044
Mathematics Subject Classification (2000): 15A60,15A72
ISBN3-540-22887-X Springer Berlin Heidelberg New York
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it specially suitable as a textbook for tensor courses. This material combines
classic matrix techniques together with novel methods and in many cases the
questions and problems are solved using different methods. They confirm the
applied orientation of the book.
A computer package, written in Mathematica, accompanies this book,
available on: In it, most of the
novel methods developed in the book have been implemented. We note that
existing general computer software packages (Mathematica, Mathlab, etc.) for
tensors are very poor, up to the point that some problems cannot be dealt
VIII Preface
with using computers because of the lack of computer programs to perform
these operations.
The main contributions of the book are:
1.
The book employs a new technique that permits one to extend (stretch)
the tensors, as one-column matrices, solve on these matrices the desired
problems, and recover the initial format of the tensor (condensation). This
technique, applied in all chapters, is described and used to solve matrix
equations in Chapter 1.
2.
An important criterion is established in Chapter 2 for all the components
of a tensor to have a given ordering, by the definition of a unique canonical
tensor basis. This permits the mentioned technique to be applied.
3.
In Chapter 3, factors are illustrated that have led to an important con-
fusion in tensor books due to inadequate notation of tensors or tensor
operations.
4.
In addition to dealing with the classical topics of tensor books, new tensor
concepts are introduced, such as the rotation of tensors, the transposer

September 30, 2004 Enrique Castillo
Contents
Part I Basic Tensor Algebra
Tensor Spaces 3
1.1 Introduction 3
1.2 Dual or reciprocal coordinate frames in affine Euclidean spaces 3
1.3 Different types of matrix products 8
1.3.1 Definitions 8
1.3.2 Properties concerning general matrices 10
1.3.3 Properties concerning square matrices 11
1.3.4 Properties concerning eigenvalues and eigenvectors 12
1.3.5 Properties concerning the Schur product 13
1.3.6 Extension and condensation of matrices 13
1.3.7 Some important matrix equations 17
1.4 Special tensors 26
1.5 Exercises . 30
Introduction to Tensors 33
2.1 Introduction 33
2.2 The triple tensor product linear space 33
2.3 Einstein's summation convention 36
2.4 Tensor analytical representation 37
2.5 Tensor product axiomatic properties 38
2.6 Generalization 40
2.7 Illustrative examples , 41
2.8 Exercises 46
Homogeneous Tensors 47
3.1 Introduction 47
3.2 The concept of homogeneous tensors 47
3.3 General rules about tensor notation 48
3.4 The tensor product of tensors 50

5.5 Criteria for tensor character based on contraction 122
5.6 The contracted tensor product in the reverse sense: The
quotient law 124
5.7 Matrix representation of permutation homomorphisms 127
5.7.1 Permutation matrix tensor product types in K"^ . . . 127
5.7.2 Linear span of precedent types 129
5.7.3 The isomers of a tensor 137
5.8 Matrices associated with simply contraction homomorphisms . 141
5.8.1 Mixed tensors of second order (r = 2): Matrices 141
5.8.2 Mixed tensors of third order (r = 3) 141
5.8.3 Mixed tensors of fourth order (r = 4) 142
5.8.4 Mixed tensors of fifth order (r = 5) 143
5.9 Matrices associated with doubly contracted homomorphisms . 144
5.9.1 Mixed tensors of fourth order (r = 4) 144
Contents XI
5.9.2 Mixed tensors of fifth order (r = 5) 145
5.10 Eigentensors 159
5.11 Generahzed multihnear mappings 165
5.11.1
Theorems of simihtude with tensor mappings 167
5.11.2
Tensor mapping types 168
5.11.3
Direct n-dimensional tensor endomorphisms 169
5.12 Exercises 183
Part II Special Tensors
6 Symmetric Homogeneous Tensors: Tensor Algebras 189
6.1 Introduction 189
6.2 Symmetric systems of scalar components 189
6.2.1 Symmetric systems with respect to an index subset 190

7.3.1 Number of strict components of an anti-symmetric
system with respect to an index subset 229
7.3.2 Number of strict components of an anti-symmetric
system 229
7.4 Tensors with anti-symmetries: Tensors with branched
anti-symmetry; anti-symmetric tensors 230
7.4.1 Generation of anti-symmetric tensors 232
7.4.2 Intrinsic character of tensor anti-symmetry:
Fundamental theorem of tensors with anti-symmetry . 236
7.4.3 Anti-symmetric tensor spaces and subspaces. Vector
subspaces associated with strict components 243
7.5 Anti-symmetric tensors from the tensor algebra perspective . . 246
7.5.1 Anti-symmetrized tensor associated with an
arbitrary pure tensor 249
7.5.2 Extension of the anti-symmetrized tensor concept
associated with a mixed tensor 249
7.6 Anti-symmetric tensor algebras: The ^H product 252
7.7 Illustrative examples 253
7.8 Exercises 265
8 Pseudotensors; Modular, Relative or Weighted Tensors 269
8.1 Introduction 269
8.2 Previous concepts of modular tensor establishment 269
8.2.1 Relative modulus of a change-of-basis 269
8.2.2 Oriented vector space 270
8.2.3 Weight tensor 270
8.3 Axiomatic properties for the modular tensor concept 270
8.4 Modular tensor characteristics 271
8.4.1 Equality of modular tensors 272
8.4.2 Classification and special denominations 272
8.5 Remarks on modular tensor operations: Consequences 272

9.3.1 Addition and multiplication by an scalar 324
9.3.2 Generalized exterior tensor product: Exterior
product of exterior vectors 325
9.3.3 Anti-commutativity of the exterior product /\ 331
9.4 Dual exterior algebras over V^{K) spaces 331
9.4.1 Exterior product of r linear forms over V^{K) 332
9.4.2 Axiomatic tensor operations in dual exterior
Algebras
/\^^{K).
Dual exterior tensor product 333
9.4.3 Observation about bases of primary and dual
exterior spaces 334
9.5 The change-of-basis in exterior algebras 337
9.5.1 Strict tensor relationships for /\^\K) algebras 338
9.5.2 Strict tensor relationships for An* (^) cilgebras 339
9.6 Complements of contramodular and comodular scalars 341
9.7 Comparative tables of algebra correspondences 342
9.8 Scalar mappings: Exterior contractions 342
9.9 Exterior vector mappings: Exterior homomorphisms 345
9.9.1 Direct exterior endomorphism 350
9.10 Exercises 383
10 Mixed Exterior Algebras 387
10.1 Introduction 387
10.1.1 Mixed anti-symmetric tensor spaces and their strict
tensor components 387
10.1.2 Mixed exterior product of four vectors 390
10.2 Decomposable mixed exterior vectors 394
10.3 Mixed exterior algebras: Terminology 397
10.3.1 Exterior basis of a mixed exterior algebra 397
10.3.2 Axiomatic tensor operations in the /\^ {K) algebra 398

11.10.3 Tensor product of Euclidean (pseudo-Euclidean)
tensors 452
11.10.4 Euclidean (pseudo-Euclidean) tensor contraction 452
11.10.5 Contracted tensor product of Euclidean or
pseudo-Euclidean tensors 455
11.10.6 Euclidean contraction of tensors of order r = 2 457
11.10.7 Euclidean contraction of tensors of order r = 3 457
11.10.8 Euclidean contraction of tensors of order r = 4 457
11.10.9 Euclidean contraction of indices by the Hadamard
product 458
11.11 Euclidean tensor metrics 482
11.11.1 Inner connection 483
11.11.2 The induced fundamental metric tensor 484
Contents XV
11.11.3 Reciprocal and orthonormal basis 486
11.12 Exercises 504
12 Modular Tensors over £J^(]R) Euclidean Spaces 511
12.1 Introduction 511
12.2 Diverse cases of linear space connections 511
12.3 Tensor character of y/\G\ 512
12.4 The orientation tensor: Definition 514
12.5 Tensor character of the orientation tensor 514
12.6 Orientation tensors as associated Euclidean tensors 515
12.7 Dual or polar tensors over E'"'(]R) Euclidean spaces 516
12.8 Exercises 525
13 Euclidean Exterior Algebra 529
13.1 Introduction 529
13.2 Euclidean exterior algebra of order r = 2 529
13.3 Euclidean exterior algebra of order r (2 < r < n) 532
13.4 Euclidean exterior algebra of order r=n 535

Bibliography 659
Index 663
Part I
Basic Tensor Algebra
Tensor Spaces
1.1 Introduction
In this chapter we give some concepts that are required in the remaining chap-
ters of the book. This includes the concepts of reciprocal coordinate frames,
contravariant and covariant coordinates of a vector, some formulas for changes
of basis, etc.
We also introduce different types of matrix products, such as the ordinary,
the tensor or the Schur products, together with their main properties, that
will be used extensively to operate and simplify long expressions throughout
this book.
Since we extend and condense tensors very frequently, i.e., we represent
tensors as vectors to take full advantage of vector theory and tools, and then
we recover their initial tensor representation, we present the corresponding
extension and condensation operators that permit moving from one of these
representations to the other, and vice versa.
These operators are used initially to solve some important matrix equa-
tions that are introduced, together with some interesting applications.
Finally, the chapter ends with a section devoted to special tensors that are
used to solve important physical problems.
1.2 Dual or reciprocal coordinate frames in affine
Euclidean spaces
Let £"^(11) be an n-dimensional affine linear space over the field IR equipped
with an inner connection (inner or dot product), < •,
•
>, and let {e^} be
a basis of E'"'(IR). The vector V with components {x^} in the initial basis

written as
V=\\ea\\Xn,l
and also as
V = \\ii\\Xn,i,
where the initial Xn,i and new Xn,i components are related by
Xn,l = CXn,l. (1.5)
It is obvious that any change-of-basis can be performed with the only
constraint of having an associated C non-singular matrix (|C| 7^ 0).
However, there exists a very special change-of-basis that is associated with
the matrix G
C~G-\ (1.6)
for which the resulting new basis will not be denoted by {e^}, but with the
special notation
{e*^^},
and it will be called the reciprocal or dual basis of
{ea}'
The vector y = Jleajl-X with components {x^} in the initial basis now has
the components
{XQ,},
that is
1.2 Dual or reciprocal coordinate frames in affine Euclidean spaces
V
r
_>*1 _,*2
e e
Xi
X2
Hence, taking into account (1.6), expression (1.5) leads to
X = G-^X* 4^ X* - GX
and from (1.4) we get

sion (1.2):
<V^W> = X*G(G-^F*) = XV* = XVy* (1.9)
The surprising result is that with data vectors in contra-covariant coordi-
nates the heterogeneous connection matrix is the identity matrix /, and
the result can be obtained by a direct product of the data coordinates.
From this result, one can begin to understand that the simultaneous use
of data in contra and cova forms can greatly facilitate tensor operations.
3.
If y is given in covariant coordinates (X*) and W in contravariant coor-
dinates (matrix y), proceeding in a similar way with vector V, and using
(1.7),
one gets
6 1 Tensor Spaces
< V,W>={G-^X*YGY
=
{X*)\G-'^YGY
=
{X*YG-'^GY
=
{X*flY,
(1.10)
where once more we observe that cova-contravariant data imply a unit
connection matrix /.
4.
Finally, if one has cova-covariant data, that is, V{X*) and
1^(1^*),
the
result will be
<V,W> - X^GY - (G-iX*)*G(G-^y*)
=

• Sj — Cj •
Ci
— Qji we get G — G^.
If in the linear space we consider the change-of-basis
||e^||
= ||e4||C, then
we have
G
• \\en
|e-||C)*.(||e,||C)=C*(i|e,
• e,:
\)C
and using (1.13), we finally get G = C^GC, which is the desired result.
Next, an example is given to clarify the above material.
D
Example 1.2 (Linear operator and scalar product). Assume that in the affine
linear space E'^(]R) referred to the basis {ca}^ a given linear operator (of
associated matrix T given in the cited basis) transforms the vectors in the
affine linear space into vectors of the same space. In this situation, one per-
forms a change-of-basis in E'^CSl) (with given associated matrix C). We are
interested in finding the matrix M associated with the linear operator, such
that taking vectors in contravariant coordinates of the initial basis returns the
transformed vectors in "covariant coordinates" of the new basis.
We have the following well-known relations:
1.2 Dual or reciprocal coordinate frames in affine Euclidean spaces 7
1.
In the initial frame of reference, when changing bases, the Gram matrix
(see the change-of-basis for the bilinear forms) satisfies
G = C^GC. (1.14)
2.

• • •
+ p^^e,, (1.18)
where
[g'^^]
= G
-^
is the inverse of G and symmetric, and then
IGI'
9'' = TTTT, (1-19)
where G*-^ is the adjoint of ^^j in G.
The modulus of the vector e"*^ is
V"
< e
*%
e
*^
> = v^ = 1/ T7^'
(^-^O)
which gives the scales of each main direction OX^, in the reciprocal system,
which are the reciprocal of the scales of the fundamental system (contravari-
ant) when G is diagonal.
Since <
e**%
Cj
> =
0;
Vf 7^
j,
all e*^ has a direction that is orthogonal to
all remaining vectors ej {j ^ i). All this recalls the properties of the "polar

= '^aiabaj] z
==
1,
2, ,
m; j = l,2, ,n.
Definition 1.2 (External product of matrices). Consider the following
matrices:
^m,h = L^ÜJ' ^m,h = \Pij\')
and the scalar X e K. We say that the matrix P is the external product of the
scalar X and the matrix A, and is denoted by Xo A, iff
P = X o A
=^
Pij = Xüij.
Definition 1.3 (Kronecker, direct or tensor product of matrices).
Consider the following matrices:
^m,n — [<^a/3j5 •'-^P,Q — L^7<5j? ^mp,nq — \Pii
1.3 Different types of matrix products
We say that the matrix P is the Kronecker, direct or tensor product of matrices
A and B and it is denoted by A ^ B, iff
P = A^B^Pij = aaßb^s = a[_i^j+i,[_i:^j+iöi_Li^jpj_L2^jg, (1.22)
where z =
1,2, ,
mp,
;
j =
1,2, ,
nq, [x\ is the integer part of x, with an
order fixed by convention and represented by means of ''submatrices'\'
P — Ä
>B

1;
j = i-
P
P\
J-
(1.23)
1
(1.24)
Some authors call this product the total product of matrices, which causes
confusion with the total product of linear spaces.
Definition 1.4 (Hadamard or Schur product of matrices). Consider
the following matrices:
We say that the matrix P is the Hadamard or Schur product of matrices A
and B, and it is denoted by AUB, iff
^m,n — -^m.n^-t^m^n ^ Pij —
(^ij^ij'i
^ — I5 -^5 • • •
5
^^5 J — 1, ^, . . . , ?2.
10 1 Tensor Spaces
1.3.2 Properties concerning general matrices
The properties of the sum + and the ordinary product • of matrices, which
are perfectly developed in the linear algebra of matrices, are not developed
here.
Conversely, the most important properties of other products are given.
The most important properties of these products for general matrices are:
1.
A<^{B ^C)
=^
{A^B)<S)C (associativity of (g)).

'<y
•'^q,s) ^^ ^mp^nq • ^nq,rs
In addition, we have
A- • 0 = Arm^ri • ^n,r ^^ ^m.r
and
— J^mp,rs'
B • F =
Bp^q
•
Fq^s
= Qp,s
and then
(A • C) (g) (P • F) = P^,, 0
Q;,,
= Rmp,rs.
where these formulas aim only to justify the dimensions of the data ma-
trices.
6. Generalization of the relations between scalar and tensor products:
(Ai(g)Pi)«(A2(g)P2)*-
•
-{Ak^Bk) = {Ai*A2-
•
'•Ak)^{BimB2*-
-—Bk).
This is how one moves from several tensor products to a single one. This
is possible only when the dimensions of the corresponding matrices allow
the inner product.
7.
There is another way of generalizing Property 5, which follows.
Consider now the product

Ci) • (A2
(8)
^2
(S)
C2) = (Al • A2)
(8)
(Bi • B2)
(8)
(Ci • C2),
which after generalization leads to
(Ai(8)A2(8)-
•
•(8)Afc)»(5i(8)^2(8)-
•
-(8)5^) = (Ai#5i)(8)(A2*B2)g)-
•
^0{Ak^Bk).
8. If we denote by A^ the product A • A •
• • •
• A and by A^^^ the product
A0A0 '(S)A,
with /c G IN, we have
A^,n,Bn,,^(A.5)[^^=At^UBW.
We remind the reader that {A*B)^ ^ A^ • B^, unless A and B commute.
9.
rank (A 0 5) = (rank A) (rank B)
= (rank 5) (rank A)
= rank {B 0 A). (1.25)
1.3.3 Properties concerning square matrices
Next we consider only square matrices, that is, of the form Am,m and Bp^p.

Tensor Spaces
1.3.4 Properties concerning eigenvalues
and
eigenvectors
Let
{Xi\i — 1,2, , m} and
{[ii\i
= 1,2, ,p} be the
sets
of
eigenvalues
of
^m,m^
and
Bp^p^ respectively.
If Vi
(column matrix)
is an
eigenvector
of Am^
of eigenvalue A^
and Wj
(column matrix)
is an
eigenvector
of Bp^ of
eigenvalue
/iy, that
is, if Am *
Vi

m; j = 1,2,
,p}. (1.27)
Remark
1.2. The
matrix
A can be
replaced
by the
matrix
A^ and the
matrix
B by the
matrix
B^. D
3.
The set of
eigenvectors
of the
matrix
A<^ B is the set
{vi®Wj\i
=
1,2, ,m-,
j =
1,2, ,p}.
Proof.
{A(S)B)*{vi0Wj)
=
{A9Vi)0{B»Wj)
=

the
Laplace discrete bidimensional matrix.
Since
the
eigenvalues
of
matrix
An,n are
2
-
-1
0-
0
0
-1
2
-
-1
0
0
0
-1
2
-
0
0
0-
0-
-1
•

A = B = A^^n^
according
to the
Property
2
above,
the set of
eigenvalues
of L^2 ,^2 is
{Ae,}
^ 4
sm
TTl
2(n
+ l)
+
sin
TTJ
2(n4-l)
;
ij = 1,2, ,n
D