This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.

1. .
Introduction to Tensors
PFI Seminar
Kenta OONO
2012/07/11
Kenta OONO
Introduction to Tensors

2. Self Introduction
• Kenta OONO (Twitter:@delta2323 )
• PSS(Professional and Support Service) Division
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• 2010 PFI Summer Intern → 2011 Engineer
Kenta OONO
Introduction to Tensors

3. Agenda
• What is Tensor?
• Decomposition of Matrices and Tensors
• Symmetry Parametrized by Young Diagram
Kenta OONO
Introduction to Tensors

4. Agenda
• What is Tensor?
• Decomposition of Matrices and Tensors
• Symmetry Parametrized by Young Diagram
Kenta OONO
Introduction to Tensors

5. Vector as a Special Case of Tensor
Consider a n dimensional vector (Below is the case n = 3)
 
7
 
v = 4 (1)
9
When we write this vector as v = (vi ), we have v1 = 7, v2 = 4,
and v3 = 9. As v has One index, this vector is regarded as Rank 1
Tensor.
Note
• We use 1-index notation, not 0-index.
• We somtimes use rank in a different meaning, (also in the
context of tensors).
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Introduction to Tensors

6. Matrix as a Special Case of Tensor
Consider a n × n Matrix (Below is the case n = 3).
 
7 10 3
 
A = 4 −1 −2 (2)
9 4 5
We write this matrix as A = (aij )1≤i,j≤3 .
• e.g. a11 = 7, a23 = −2, a32 = 4.
A has Two indices (namely i and j) and these indeices run through
1 to 3. A can be regarded as Rank 2 Tensor.
→ What happens if the number of indices are not just 1 or 2 ?
Kenta OONO
Introduction to Tensors

7. An Example of Rank 3 Tensor
Consider the set of real numbers X = (xijk )1≤i,j,k≤3 . As i, j, k run
throuth 1 to 3 independently, this set contains 3 × 3 × 3 = 27
elements.
If we fix the value of i, we can write this set in a matrix form.
     
7 10 3 10 2 1 5 −10 18
     
x1∗∗ = 4 −1 −2 , x2∗∗ =  3 −1 0 , x3∗∗ = −4 1 −2 
9 4 5 8 1 4 9 −4 −10
(3)
Note
• In some field, we concatenate these matrix and name it X(1)
• This is called Flattening or Matricing of X .
Kenta OONO
Introduction to Tensors

8. Is This a Multidimensional Array?
We can realize a tensor as a multiple dimensional array in most
programming languages.
PseudoCode(C like):
int[3][3][3] x;
x[1][1][2] = -1;
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Introduction to Tensors

9. Inner Product of Tensors
Let X = (xijk ), Y = (yijk ) be two rank 3 tensors and G = (g ij ) be
a symmetric (i.e. g ij = g ji ) and positive definite matrix. We can
introduce an inner product of X and Y by:
∑
n
< X , Y >= g ai g bj g ck xabc yijk (4)
a,b,c,i,j,k=1
Note:
• We can similarly define an inner product of two arbitrary rank
tensor
• X and Y must have same rank.
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Introduction to Tensors

10. An Example of Inner Product (Vector)
Let’s consider the case of rank 1 tensor ( = vector). We can write
X = (xi ), Y = (yi ). And we set G = (g ij ) an identity matrix:
 
1 ··· 0
. .
G = . 1
. .
. (5)
0 ··· 1
In this case, this definition of inner product reduce to ordinal inner
product of two vectors, namely,
∑ ∑
< X , Y >= g ai xa yi = xi yi (6)
a,i i
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Introduction to Tensors

11. An Example of Tensor
Suppose we have a (smooth) function f : R3 → R. We can derive
tensors of arbitrary rank from this function.
Taking Gradient, we obtain rank 1 tensor. (Please replace (1, 2,
3) with (x , y , z) and vice versa.
 ∂f 
 ∂x 
∇f = (∇i f ) =  ∂y 
∂f
(7)
∂f
∂z
Similarly, Hessian is an example of rank 2 tensor.
 
∂2f ∂2f ∂2f
 ∂x2∂x ∂x ∂y ∂x ∂z 
H(f ) = (Hij (f )) =  ∂ f ∂2f ∂2f 
(8)
 ∂y ∂x ∂y ∂y ∂y ∂z 
∂2f ∂2f ∂2f
∂z∂x ∂z∂y ∂z∂z
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Introduction to Tensors

12. Supersymmetry Property
By taking derivative, we can create an arbitrary rank of tensors.
These tensors are (Super)Symmetric in the sense that
interchange of any two indices remains the tensor identical. For
example,
fx y z = fx zy = fy x z = fy zx = fzx y = fzy x . (9)
Note:
• Afterward, we write this symmetry as .
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Introduction to Tensors

13. Agenda
• What is Tensor?
• Decomposition of Matrices and Tensors
• Symmetry Parametrized by Young Diagram
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Introduction to Tensors

14. Decompose!
It is fundamental in mathematics to decompose some object into
simpler, easier-to-handle objects.
∑ √
• Fourier Expansion : f = an exp( −1nx )
n∈Z
∑
• Legendre Polynomial : P = ak Pk
k
1 dk 2
• where Pk (x ) = (x − 1)k
k!2k dx k
• Jordan-H¨lder : e = G0 ◁ G1 ◁ G2 ◁ · · · ◁ Gn−1 ◁ Gn = G
o
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Introduction to Tensors

15. Decomposition of Matrix
There are many theories regarding the decomposition (or,
factorization) of matrices.
• SVD
• LU
• QR
• Cholesky
• Jordan
• Diagonalization ...
Although we have theories of factorization of Tensors (e.g. Tucker
Decomposition, higher-order SVD etc.) We do NOT go this
direction. Instead, we consider decomposition of matrix into
Summation of matrix.
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Introduction to Tensors

16. Example: Projection to Axis
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Introduction to Tensors

17. Example of Decomposition of Matrix
We can decompose matrix into Symmetric part and
Antisymmetric part.
Example:
     
7 10 3 7 7 6 0 3 −3
     
4 −1 −2 = 7 −1 1 + −3 0 −3
9 4 5 6 1 5 3 3 0
Symmetric Part Antisymmetric Part
A Asym Aanti
(10)
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Introduction to Tensors

18. How to Calculate Sym. and Antisym. Part
We can calculate the symmetic and antisymmetric part by simple
calculation (Exercise!).
   
7 10 3 7 4 9
1    
Asym = 4 −1 −2 + 10 −1 4
2
9 4 5 3 −2 5 (11)
sym 1
aij = (aij + aji )
2
   
7 10 3 7 4 9
1    
Aanti = 4 −1 −2 − 10 −1 4
2
9 4 5 3 −2 5 (12)
1
anti
aij = (aij − aji )
2
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Introduction to Tensors

19. Picture of Projection to Sym. and Antisym. Part
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Introduction to Tensors

20. Notation Using Young Diagram
We can symbolize symmetry and antisymmetry with Young
Diagram
     
7 10 3 7 7 6 0 3 −3
     
4 −1 −2 = 7 −1 1 + −3 0 −3
9 4 5 6 1 5 3 3 0 (13)
A Asym Aanti
The number of boxes indicates rank of the tensor (one box for
vector and two boxes for matrix).
Kenta OONO
Introduction to Tensors

21. Agenda
• What is Tensor?
• Decomposition of Matrices and Tensors
• Symmetry Parametrized by Young Diagram
Note:
• From now on, we concentrate on Rank 3 Tensors (i.e.
k = 3).
• And we assume that n = 3, that is, indices run from 1 to 3.
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Introduction to Tensors

22. Symmetrizing Operator
We consider the transformation of tensor:
T : X = (xijk ) → X ′ = (xijk ),
′
(14)
′
where xijk = 1 (xijk + xikj + xjik + xjki + xkij + xkji )
6 (15)
= x(ijk) .
Kenta OONO
Introduction to Tensors

23. Young Diagram and Symmetry of Tensor (Sym. Case)
Let X = (x123 ) be a tensor of rank 3, we call X Has a Symmetry
of , if interchange of any of two indices doesn’t change each
entry of X .
Example:
• If X has a symmetry of , then
x112 = x121 = x211
(16)
x123 = x132 = x213 = x231 = x312 = x321 etc...
• Symmetric matrices have a symmetry of
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Introduction to Tensors

24. Property of T
• For all X , T (X ) has symmetry of .
• If X has symmetry of , then T (X ) = X .
• For all X , T (T (X )) = T (X ).
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Introduction to Tensors

25. Antisymmetrizing Operator
Next, we consider Antisymmetrization of tensors:
T : X = (xijk ) → X ′ = (xijk ),
′
(17)
′
where xijk = 1 (xijk − xikj − xjik + xjki + xkij − xkji )
6 (18)
= x[ijk] .
Kenta OONO
Introduction to Tensors

26. Young Diagram and Symmetry of Tensor (Antisym. Case)
Let X = (x123 ) be a tensor of rank k, we call X has a symmetry of
if interchange of any of two indices change only the sign of
each entry of X .
Example.
• If X has a symmetry of , then
• x123 = −x132 = −x213 = x231 = x312 = −x321
• x112 = 0, because x112 must be equal to −x112 .
• Antisymmetric matrix has a symmetry of
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Introduction to Tensors

27. Orthogonality of Projection
T and T are orthogonal in the sense
T ◦ T (X ) = T ◦ T (X ) = 0. (19)
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Introduction to Tensors

28. Counting Dimension
How many dimension do the vector space consisting of tensors ...
• with symmetry have ?
• with symmetry have ?
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Introduction to Tensors

29. Dimensions of each subvector spaces.
If n = 3 (that is, i, j and k runs through 1 to 3), then each
dimension is as follows.
But dimension of total space is 3 × 3 × 3 = 27. This is larger than
1 + 10 = 11.
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Introduction to Tensors

30. Young Diagram
Young Diagram is an arrangent of boxes of the same size so that:
• two adjacent boxes share one of their side.
• the number of boxes in each row is non-increasing.
• the number of boxes in each column is non-increasing.
Here is an example of Young Diagram with 10 boxes.
(20)
English Notation French Notation
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Introduction to Tensors

31. Decomposition of Tensor (of Rank 3)
We have three types of Young Diagram which have three boxes,
namely,
(21)
, , and
Symmetric Antisymmetric ???
→ What symmetry does represent?
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Introduction to Tensors

32. Coefficients of Symmetrizers
We list coneffients of each operator in a tabular form.
Coefficient of xijk
ijk ikj jik jki kij kji
1/6 1/6 1/6 1/6 1/6 1/6
1/6 -1/6 -1/6 1/6 1/6 -1/6
Sum 1
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Introduction to Tensors

33. Coefficients of Symmetrizers
Coefficient of xijk
ijk ikj jik jki kij kji
1/6 1/6 1/6 1/6 1/6 1/6
1/6 -1/6 -1/6 1/6 1/6 -1/6
4/6 -2/6 -2/6
Sum 1
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Introduction to Tensors

34. Symmetrizer T
We consider the transformation of tensor:
T : X = (xijk ) → X ′ = (xijk ),
′
(22)
′
where xijk = 1
(2xijk − xjki − xkij ).
3
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Introduction to Tensors

35. Property of T
• For all X , T (X ) has symmetry of .
• If X has symmetry of , then T (X ) = X .
( )
• For all X , T T (X ) = T (X ).
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Introduction to Tensors

36. Tensors with Symmetry of
Let X = (xijk ) be a tensor of rank 3. We say X Has a Symmetry
of , if
1
xijk = (2xijk − xjki − xkij ) (23)
3
for all 1 ≤ i, j, k ≤ n.
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Introduction to Tensors

37. Where This Equation ”Symmetric” ?
Remember that X has a symmetry of , if
1
xijk = (2xijk − xjki − xkij ) (24)
3
for all 1 ≤ i, j, k ≤ n.
This equation can be transormed as follows :
1
(2xijk − xjki − xkij )
xijk =
3 (25)
⇔ 3xijk = 2xijk − xjki − xkij
⇔ xijk + xjki + xkij = 0
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Introduction to Tensors

38. Projection to Each Symmetrized Tensors
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Introduction to Tensors

39. Decomposition of Higher Rank Tensors
Higher rank tensors are also decompose into symmetric tensors
parametrized by Young Diagram.
⊕ ⊕
Rn×n×n×n×n =
⊕ ⊕ ⊕ ⊕
(26)
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Introduction to Tensors

40. Summary
• Tensors as a Generalization of Vectors and Matrices.
• Decomposition of Matrices and Tensors
• Symmetry of Tensors Parametrized by Young Diagrams.
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Introduction to Tensors

41. References
• T G. Kolda and B W. Bader, Tensor Decompositions and
Applications
• 2009, SIAM Review Vol. 51, No 3, pp 455-500
• D S. Watkins, Fundamentals of Matrix Computations 3rd. ed.
• 2010, A Wiley Series of Texts, Monographs and Tracts
• W. Fulton and J Harris, Representation Theory
• 1991, Graduate Texts in Mathematics Readins in Mathematics.
• And Google ”Representation Theory”, ”Symmetric Group”,
”Young Diagram” and so on.
Kenta OONO
Introduction to Tensors