Tensor of a quaternion

This redirect is about a historical usage of the term tensor. For the modern usage, see Tensor.

In mathematics the term tensor was introduced by Sir William Rowan Hamilton, who used it to very specifically denote the positive square root of the norm of a quanternion. This usage is distinct from the wider meaning of tensor in modern mathematics, which grew out of generalising the norm operation to more general multilinear maps.

Definition of the tensor of a quaterion

In the nomenclature of Hamiltons quaternion calculus, the word tensor can be used in two different but related contexts. The first context is as an operator, as in take the tensor of something, and the second context is as a positive or more correctly unsigned number. As Hamilton put it:

• it seems convenient to enlarge by definition the signification of the new word tensor, so as to render it capable of including also those other cases in which we operate on a line by diminishing instead of increasing its length ; and generally by altering that length in any definite ratio. We shall thus (as was hinted at the end of the article in question) have fractional and even incommensurable tensors, which will simply be numerical multipliers, and will all be positive or (to speak more properly) SignLess Numbers, that is, unclothed with the algebraical signs of positive and negative ; because, in the operation here considered, we abstract from the directions (as well as from the situations) of the lines which are compared or operated on.

The tensor of a positive scalar is the scalar itself. The tensor of a negative scalar is the scalar with out the negative sign. For example:

$\mathbf {T} (5)=5\,$

$T(-5)=5\,$

The tensor of a vector is by definition the length of the vector. For example if:

$\alpha =xi+yj+zk\,$

Then

$\mathbf {T} \alpha ={\sqrt {x^{2}+y^{2}+z^{2}}}$

A quaternion is by definition the quotient of two vectors and the tensor of a quaternion is by definition the quotient of the tensors of these two vectors. In symbols:

$q={\frac {\alpha }{\beta }}.$

$\mathbf {T} q={\frac {\mathbf {T} \alpha }{\mathbf {T} \beta }}.$

Bitensors

If Q is a biquaternion then the operation of taking the tensor of a biquaternion returns a bitensor.

$\mathbf {T} Q=t+{\sqrt {-1}}t'$

Here t and t' are reals.

Properties of Tensors

The tensor of a unit vector is equal to one, if $\alpha$ is a unit vector then:

$\mathbf {T} \alpha =1$

It is generally true for any vector or quaternion that:

$\mathbf {TU} \alpha =1$

$\mathbf {TU} q=1$

Hamilton proved that the tensor of a quaternion is equal to the square root of the common norm. In symbols this can be written:

$\mathbf {T} q={\sqrt {\mathbf {N} q}}={\sqrt {q\mathbf {k} q}}$

Hamilton also proved that if q is written as

$q=w+xi+yj+zk\,$

then

$\mathbf {T} q={\sqrt {w^{2}+x^{2}+y^{2}=z^{2}}}$

Applications

Stresses and Strains

Since the tensor of a quaternion represents its stretching factor one of its many applications is in the computations of stresses and strains.