[complex-number] Complex numbers.
Addition is the same as in . Multiplication uses or and the distributive law
Origin of complex numbers or
- Characteristic equation of the harmonic oscillator ODE
-
Another motivation for complex numbers comes from polynomial factorization. A real polynomial can be factored completely into products of the form or , while the latter can be factored in as , in particular, . So for convenience, one may choose to use complex numbers.
You can still choose to think of this as merely an algebraic convenience, without needing the geometry of complex numbers
However, in the split complex below, cannot be factored into linear polynomials. Even has four roots, besides , there are two more roots
Example [split-complex-number] Split-complex numbers.
Addition is the same as in . Multiplication uses or and the distributive law
Also see Intuitive explanation of complex numbers for the relation between multiplication by unit complex numbers and rotations in
[normed-division-algebra]
has a quadratic form , a multiplication , and for elements with unit (quadratic form) distance , the multiplication also has unit distance
Combining unit distance and scalar multiplication, this property can be expressed as
- corresponds to
by
- corresponds to
by
null elements have no multiplicative inverse
[quaternion]
Starting from or as , add a new imaginary unit
-
Define another imaginary unit
-
Different imaginary units anti-commute
-
Conjugate of an imaginary unit gives its negative
This results in
- Imaginary unit multiplication is associative
- Satisfies norm multiplication
- If starting from , , thus
- If starting from , , thus
-
give
-
give
Eaxmple [octonion] Using a new imaginary unit in (where )
Define other imaginary units
Different imaginary units anticommute
Different imaginary units anti-associate if
Conjugate of imaginary unit is negative
This makes
- norm multiplication
Similarly, according to and gives octonion for or split octonion for
-
give
-
give
What results from and the associativity of imaginary units is another algebra , which does not satisfy
[imaginary-automorphism] The method of constructing new imaginary units is not coordinate-free, so we need to consider automorphisms of imaginary units with . Since it preserves multiplication, it automatically preserves distance
Example for itโs symmetric
Question (ref-21, p.35) (ref-22, p.85)
- for
- for .
as automorphism of illustrates that, without additional structure, such as multiplication and , only the bare linear space structure cannot yield special groups like . (Although it is said that all compact groups can have matrix representations.)
[problem-of-quaternionic-linear]
Attempting to define linear algebra for . We immediately encounter a problem: one definition of a linear map is as a homomorphism between linear spaces, but since is non-commutative, scalar multiplication cannot arbitrarily commute with matrix multiplication on the same side , thus matrix multiplication on is not a linear transformation.
Therefore, the โlinear structureโ of is defined as, for example, left matrix multiplication acting as the linear map , and right scalar multiplication , such that the linear map is a homomorphism of the linear structure .