Tensor of a quaternion

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.

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:

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

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

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.

Relation to norm

In mathematics, some thinkers believe there is a relationship between the norm of a quaternion and the tensor of a quaternion. Some writers define the norm of a quaternion as having the same formula as the tensor of a quaternion, while other writers define the norm of a quaternion as the square of the tensor. Hamilton uses the term tensor in two different sences as a positive numerical quantity and as an operator that operates on other mathematical entities extracting a tensor quantity from them.

Hamilton did not, as now claimed, define a tensor to be "a signless number"; what he actually says is:

• 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.

Hamilton defined the new word tensor as a positive or more properly signless number. The tensor of a quaternion is a number which represents its magnitude, the "stretching factor", the amount by which the application of the quaternion lengthens a quantity; specifically, the tensor is defined as the square root of the norm — this is a one-dimensional quantity, quite distinct from the modern sense of tensor, coined by Woldemar Voigt in 1898 to express the work of Riemann and Ricci. As a square root, tensors cannot be negative, and the only quaternion to have a zero tensor is the zero quaternion. Since tensors are numbers, they can be added, multiplied, and divided. The tensor of the product of two quaternions is the product of their tensors; the tensor of a quotient (of non-zero quaternions) is the quotient of their tensors; but the tensor of the sum of two quaternions ranges between the sum of their tensors (for parallel quaternions) and the difference (for anti-parallel ones) .

The tensor of the quaternion q is denoted Tq.