Tensortlgebraldexing

Tensor IEIR"×↳i×h

has elements

I= Ctiase;)We can write i→=a;iz .

. .in to denote

an indexvector_ I 'E[ I.

]×[ b) x. " x[ In ]

,

where [ in ] = { 1,2,

},

...

,n }

.

The

fibresare denoted Finis

.in . :inn .

.. in

EIHEII.EE#iIEtEEtIEEIFitiisiz tte; :c ; this :

The slicesare denoted

Tiiiijjijmiiikitkti .' in

E¥#E##EIITii :

T.is :T

: :c ;

For 3- way tensors,

the frontal slice

T: :i

,

has a special short . hand notation

Iii Ti,

AtatriczutionA tensor I can be matted or

infolded or flattened in a matrix

by reordering its elements. Mode

matricizaho_ takes the mode - in fibresand stacks them as the columns of

the new matrix. If Ielpixkxixk

,

its mode -1 maotrnization is a matrix

Ten ,EIR " " 's '

s" H

,

and in general ,

the mode. nmafrieizationis q matrix

Ten ,C- lRh×lhk" ' l

n .nlntiitd

.

In mode - n matricizafion,

element

at finis ,. . .in ) maps to ( in , j )

,where

j=1&¥§ynlta . 7) J. ,

where

⇐ii. In.

with

An example ( next page) will

clarifythe process .

t.to#ricization_eIampfe.

4×3×2

Let IEIR with frontal slices

7 5 9of,:# and . ftp.YIED .

4 8

Now we have :

t.fi?Etia.tEID

t.tt?oti!aEi::Hk.dTn=li, ; hit it . oo ! Ya !} 's:)

Vectorisation

÷llectorization of a tensor stacks

the columns of its mode -1 matri .

cization in a column vector.

Ias above,

we have vec(I)=( 1, 2,3

,... ,24)T

.

tensortimes-9scalar,tectolmatrix,

censor]

Tensor times a scalar,

&I,

scales

every element of J by a EIR,

LI = ( at.→ ) .

The n . mode vector . tensor product- of

a tensor Ielphxlaeixt with vector FGIRL "

is

ITyu→c- 1/2"×b×i×h'×hm×' ' ' xk

,

like)i. .in

. .im . :ij§g=

.

tiniiinvin

=[(t→ii. .in

. iioni . .in )

V→

}ii. in

. niniiitvwhere e.

,

. > is the inner product-

oftwo vectors

.

The idea is to take

the inner products between all

mode . h fibres and F.

In tensor . vector products,precedence matters :

IEa→EI=CI # ⇒ in .is#xib7xna

If I is 2- way ( a matrix ),

we have

II. v→=FI and I 'xsu→=Iv→.

t.tk#IETEIf#¥IE⇒

Let IEIR" "2

with frontal slices

t.tn;Egeland . ftp.YLYID ,

and let v→= ( 5,

a)T

.

Then

t.ee#Et:YyoeEIIEFoIt:HthtItIo::Yn .

The

n.me#ixtenfdu

oftensor IEIRHH ' " t and matrix MEIR

'×his

IXN M e IR'

'' ' ' ' ' × In .n×J× In , ,

x. . . x In

(I×nM)i.

.in. . jinn .

.

.is#iti.i....irmjin

= [ ME... .in#.....iDii..in

That is,

we multiply each mode . n fibrewith M

. Equivalently,we can use

unfolding :

-

1 = Ixnm ⇐ > Sa,= MIN , .

If I is 2- way ( i. e. a matrix ),

Ta ,=I and

Ta,= It

.

So

I x. M ' MI and IX. M = MIT

The order of tensor - matrix

multiplicationsover different modes doesn't

matter : if M In,

then

IX. Axn B=I×nB×mA.

If the modes are the same and

AEIR '×kand BEIRK "

,

then

IxnA×nB=Ixn( BA ) .

4×3×2

Let IEIR with frontal slices

t.fi?g!dandta=EsYL!ID,

and let

M=kZIo:o)

Then, if I = IX. M

,we have

Sn, ,

= 1.1+2.2 t 4.3+6.4=41 and

S ,=( 41 93 145) and ss= ( 192299301 )410 930 1450 Mte 24903010

The tensor inner product-

oftwo tensors 5-

,I c- 1134×12 ' ' ' ×k is the

sum of their element . wise products ,

< I ,Is¥s§s.ie#nsaiioeiteiioivIen.sur_nor_m

The norm_ of a tensor IEIR"×↳ " h

is

the square . root of the sum of the

squares of its elements :

NIKKEI.it?ItiIsnif2This can be alternatively be defined

as

FIM114,,k= ,

117,.lk ,

llvedtllb,ortr-TT.tl,

or using any other

way to define the Euclidean lfrobeniushorn of a Vector / matrix

.

SymmetryTensor I is cubical if all of its

modes have the same dimensionality :

Iepixlxlx' . . xl

.

Cubical tensor is

Itai,

ifits elements remain constant under

any permutation of the indices.

IfIEIR

' " "

,it is symmetric if and

only if

tijkitiicjitjiiitjkittkij stage . for all i. j ,ke[ I ]

Tensor IER" " * " h

is ( hyper . ) diagonalif this ... in

¥0 only if i. =ia= . . . ← in .

If

4=1 ,

- - - In ,I is also symmetric .

If the diagonal entries are

ftp.#all 1,

tensor behaves similarlyto the identity matrix

.