Maths - Tensor Operations

To give us quantities to work with vectors can be represented by a linear combination of basis vectors, so vector 'a' could be represented by:

a = a1 e1 + a2 e2 + a3 e3 ...

similarly vector 'b', in the same coordinate system, could be represented by:

b = b1 e1 + b2 e2 + b3 e3 ...

where:

Inner Product

We have already covered the dot product '•' for vectors here. This concept can be extended to tensors.

Inner Product of Vectors

So using the above values we have:

ab = (a1 e1 + a2 e2 + a3 e3) • (b1 e1 + b2 e2 + b3 e3)

If we assume that the bases are orthogonal (mutually perpendicular), then, a dot product of a base with another will be zero except a base with itself which will be one:

e1 • e1 + e2 • e2 + e3 • e3 = 1
e1 • e2 + e2 • e3 + e3 • e1 = ... = 0

Which gives the result:

ab = a1 b1 + a2 b2 + a3 b3

Equivalent Matrix Equation

aT b

a1 a2 a3
b1
b2
b3
=
a1 b1 + a2 b2 + a3 b3

Inner Product of Covectors

ab = (a1 e1 + a2 e2 + a3 e3) • (b1 e1 + b2 e2 + b3 e3)

So what is the product covector bases?

e1 • e1 = ?
e1 • e2 = ?

Equivalent Matrix Equation

a bT

a1
a2
a3
a1 a2 a3
=
a1*b1 a2*b1 a3*b1
a1*b2 a2*b2 a3*b2
a1*b3 a2*b3 a3*b3

 

Vector times covector:

b1
b2
b3
a1 a2 a3
=
a1•b1 a2•b1 a3•b1
a1•b2 a2•b2 a3•b2
a1•b3 a2•b3 a3•b3

Tensor Product

This denoted by tensor product, it is a more general case of the Kronecker product, it takes the right hand operand and duplicates it in every element of the left hand operand. Each of these copies is scalar multiplied by the corresponding element.

kronecker

The tensor product may increase the dimension or the rank of the result depending whether the multiplicands have common or independent bases.

In the above example, both the operands are rank 2 and they both use the same bases, if the operands are 3×3 matrices the result is a 9×9 matrix. So, if we use the same bases, a matrix of dimension m by n tensor product a matrix of dimension u by v will give a resultant matrix of dimension m*u by n*v.

In the case below, the two vectors have different bases, shown as a column vector tensor product row vector which gives a matrix:

b1
b2
b3
tensor product
a1 a2 a3
=
a1*b1 a2*b1 a3*b1
a1*b2 a2*b2 a3*b2
a1*b3 a2*b3 a3*b3

So if we have independent basis vectors, say ei,ej and ek so the result will contain all the bases in all the operands, eijtensor product ek=eijk

In other words, if all the bases of the operands are independent, the rank of the result will be the sum of the ranks of the operands.

So in terms of the vectors defined at the top of the page, we have:

a tensor product b = (a1 e1 + a2 e2) tensor product (b1 e1 + b 2 e2)
a tensor product b = a1*b1 (e1tensor producte1) + a1*b2 (e1tensor producte2) + a2*b1 (e2tensor producte1) + a2*b2 (e2tensor producte2)

 

Other Tensor Products

Inner tensor product or contraction

by a vector decreases the rank of a tensor

Outer tensor product

by a vector increases the rank of tensors.

Wedge Product

Two dimentional case

As we did for the tensor product we can take the vectors defined at the top of the page, and multiply them out:

a ^ b = (a1 e1 + a2 e2) ^ (b1 e1 + b 2 e2)
a ^ b = a1*b1 (e1^e1) + a1*b2 (e1^e2) + a2*b1 (e2^e1) + a2*b2 (e2^e2)

the difference from the tensor product is that the square of a vector is zero (as cross product in 3 dimensions) which gives:

e1^e1 = e2^e2 = 0

so:

a ^ b = a1*b2 (e1^e2) + a2*b1 (e2^e1)

also it anti-commutes so:

e1^e2= - e2^e1

which allows us to simplify to:

a ^ b = (a1*b2 - a2*b1)(e1^e2)

However it won't simplify any further, in other words e1^e2 cant be defined in terms of e1and e2 separately.

Three dimentional case

As we did for the tensor product we can take the vectors defined at the top of the page, and multiply them out:

a ^ b = (a1 e1 + a2 e2+ a3 e3) ^ (b1 e1 + b 2 e2+ b 3 e3)
a ^ b = a1*b1(e1^e1) + a1*b2(e1^e2) + a1*b3 (e1^e3)
+ a2*b1(e2^e1) + a2*b2(e2^e2)+ a2*b3(e2^e3)
+ a3*b1(e3^e1) + a3*b2(e3^e2)+ a3*b3(e3^e3)

as before:

e1^e1 = e2^e2 =e3^e3 = 0

and

eu^ev= (-1)nm ev^eu

which allows us to simplify to:

a ^ b = (a1*b2- a2*b1)(e1^e2)
+ (a2*b3 - a3*b2)(e2^e3)
+ (a3*b1 -a1*b3 )(e3^e1)

as before this won't simplify any further, in other words e1^e2 , e2^e3
and e3^e1cant be defined in terms of e1, e2 and e3 separately.

Exterior Product

p-form and q-form

 

Triple Product

If

t = u • (v × w)

then in tensor notation:

t = ui eijk vj wk = eijk ui vj wk


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Book Shop - Further reading.

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cover us uk de jp fr ca Schaum's Outline of Theory and Problems of Tensor Calculus - I'm finding this hard going, it starts off with as review of linear algebra, matrix notation, etc. It redefines a lot of conventions which are hard to relearn, such as superscrips instead of subscripts to identify elements, and a summation convention, then it goes into coordinate transformations. I cant find any definition of what a tensor is. I think this book is aimed at people who already have some knowledge of the subject.

I can't find any mention in this book of the term hyper-matrix.

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