1 is the unit element, the other three elements are multiplied by-1, and the pairwise combination, the positive sequence is the third element, the reverse sequence is the positive sequence multiplied by-1, and the combination of the three sequences is also-1. In fact, from here on, I feel similar to external algebra, anti-symmetry.
Multiplication table, knowing the multiplication table, you can fully understand the combinatorial relationship of groups, and you can fully understand the algebraic structure of groups. On the diagonal, 1 is a second-order element (except the identity element 1 itself), and on the diagonal,-1 is a fourth-order element. Then there is exchangeability. If it is a symmetric matrix or table, it is a commutative group. This group obviously does not communicate. The exchanged parts are extracted to form a sub-multiplication table, which is the multiplication table of the central subgroup (the elements in the central subgroup are exchanged with all the elements in the group). The central subgroup here only contains.
The subgroups of quaternions include trivial groups, second-order cyclic groups and three fourth-order cyclic groups.
Consider a vector space, in which the elements are called quaternions, which are composed of four real numbers and can be divided into scalar parts and vector parts.
This vector space defines multiplication and becomes associative algebra of quaternions.
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Actually, it is the usual scalar product and vector product.
Historically, these two products were separated from quaternion multiplication by Gibbs by taking the scalar part of quaternion as 0.
Interesting. It used to be simple, but now it's back to complex form. Simplicity is not necessarily the essence.
Quaternion algebra constitutes a noncommutative field, with real numbers and complex numbers as its special cases. The * * * yoke of quaternion, the sum of squares of modules is defined as
And has the following properties
Then the scalar part satisfies the commutative law of multiplication. Can be extended to limited situations.
There is a display problem, which has little impact.
The inverse element is constructed by taking the product of the self-yoke as the modular square. The modulus square of the inverse element, taking the yoke of * * * just adds up to a scalar. Then there is the nature of the inverse operation. Like * * * yoke, pairs form matching pairs, generate identities, eliminate identities, and the composition of functions is similar. In fact, this is how the inversion operation is constructed. It is necessary to construct the exchange quantity from the non-exchange quantity, so as to realize the elimination of the quantity or the change of the operation order. Maybe we can look at noncommutative algebra to get more understanding.
Quaternion division is essentially solving equations or. With the help of previous knowledge about inverse elements, it is easy to get
Because multiplication is not commutative, it will produce two results. Their modules are obviously equal.
According to my understanding, it is generally impossible to write, because the result is not unique and the definition is not clear.
Algebra is vector space plus multiplication. When this multiplication is quaternion multiplication, it is quaternion algebra. This multiplication is complicated, but it can reflect some natural properties.