Classical Hamiltonian Quaternions - Other Operators in Detail - Tensor

T is the tensor operator. It returns a kind of number called a tensor.

The tensor of a positive scalar is that scalar. The tensor of a negative scalar is the absolute value of the scalar (i.e., without the negative sign). For example:

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

Then

The tensor of a unit vector is one. Since the versor of a vector is a unit vector, the tensor of the versor of any vector is always equal to unity. Symbolically:

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:

From this definition it can be shown that a useful formula for the tensor of a quaternion is:

It can also be proven from this definition that another formula to obtain the tensor of a quaternion is from the common norm, defined as the product of a quaternion and its conjugate. The square root of the common norm of a quaternion is equal to its tensor.

A useful identity is that the square of the tensor of a quaternion is equal to the tensor of the square of a quaternion, so that parenthesis may be omitted.

Also, the tensors of conjugate quaternions are equal.

If Q is a biquaternion then its tensor is a bitensor.

Here t and t′ are real numbers.