Tensor of a quaternion
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Hi every body, there needs to be an article on the concept that a quaternion has a tensor. I don't think that this discussion should be limited to only 19th century sources like the article classical hamiltonian quaternions. Clearly this is an idea that moves into the 20th century and evolves.
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.[failed verification] [1][2][3]The tensor of a quaternion is a number which represents its magnitude,[4] the "stretching factor"[5], the amount by which the application of the quaternion lengthens a quantity; specifically, the tensor is defined[citation needed] as the square root of the norm [6] — 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.[7] As a square root, tensors cannot be negative[citation needed], and the only quaternion to have a zero tensor is the zero quaternion[citation needed]. 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.
- ^ Hamilton 1853 pg 57
- ^ Hardy 1881 pg 5
- ^ Tait 1890 pg.31 explains Hamilton's older definition of a tensor as a positive number
- ^ Tait
- ^ Tait (1890), pg 32
- ^ Cayley (1890), pg 146,
- ^ OED, "Tensor", def. 2b, and citations.