Fghjjk

Γεωμετρική Άλγεβρα

Blades, grading, and canonical basis

A multivector that is the outer product of r independent vectors ($r\leq n$) is called a blade, and the blade is said to be a multivector of grade r. From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.

Consider a set of r independent vectors $\{a_{1},...,a_{r}\}$ spanning an r-dimensional subspace of the vector space. With these, we can define a real symmetric matrix

$[\mathbf {A} ]_{ij}=a_{i}\cdot a_{j}$

By the spectral theorem, A can be diagonalized to diagonal matrix D by an orthogonal matrix O via

$\sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {A} ]_{kl}[\mathbf {O} ^{\mathrm {T} }]_{lj}=\sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {O} ]_{jl}[\mathbf {A} ]_{kl}=[\mathbf {D} ]_{ij}$

Define a new set of vectors $\{e_{1},...,e_{r}\}$, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:

$e_{i}=\sum _{j}[\mathbf {O} ]_{ij}a_{j}$

Since orthogonal transformations preserve inner products, it follows that $e_{i}\cdot e_{j}=[\mathbf {D} ]_{ij}$ and thus the $\{e_{1},...,e_{r}\}$ are perpendicular. In other words the geometric product of two distinct vectors $e_{i}\neq e_{j}$ is completely specified by their outer product, or more generally

${\begin{array}{rl}e_{1}e_{2}\cdots e_{r}&=e_{1}\wedge e_{2}\wedge \cdots \wedge e_{r}\\&=\left(\sum _{j}[\mathbf {O} ]_{1j}a_{j}\right)\wedge \left(\sum _{j}[\mathbf {O} ]_{2j}a_{j}\right)\wedge \cdots \wedge \left(\sum _{j}[\mathbf {O} ]_{rj}a_{j}\right)\\&=\det[\mathbf {O} ]a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}\end{array}}$

Therefore every blade of grade r can be written as a geometric product of r vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a block matrix that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace are normalized according to

${\hat {e}}_{i}={\frac {1}{\sqrt {|e_{i}\cdot e_{i}|}}}e_{i},$

then these normalized vectors must square to +1 or −1. By Sylvester's law of inertia, the total number of +1s and the total number of −1s along the diagonal matrix is invariant. By extension, the total number p of these vectors that square to +1 and the total number q that square to −1 is invariant. (If the degenerate case is allowed, then the total number of basis vectors that square to zero is also invariant.) We denote this algebra ${\mathcal {G}}(p,q)$. For example, ${\mathcal {G}}(3,0)$ models 3D Euclidean space, ${\mathcal {G}}(1,3)$ relativistic spacetime and ${\mathcal {G}}(4,1)$ a 3D conformal geometric algebra.

The set of all possible products of n orthogonal basis vectors with indices in increasing order, including 1 as the empty product forms a basis for the entire geometric algebra (an analogue of the PBW theorem). For example, the following is a basis for the geometric algebra ${\mathcal {G}}(3,0)$:

$\{1,e_{1},e_{2},e_{3},e_{1}e_{2},e_{1}e_{3},e_{2}e_{3},e_{1}e_{2}e_{3}\}$

A basis formed this way is called a canonical basis for the geometric algebra, and any other orthogonal basis for V will produce another canonical basis. Each canonical basis consists of 2ⁿ elements. Every multivector of the geometric algebra can be expressed as a linear combination of the canonical basis elements. If the canonical basis elements are {B_i | i∈S} with S being an index set, then the geometric product of any two multivectors is

$(\Sigma _{i}\alpha _{i}B_{i})(\Sigma _{j}\beta _{j}B_{j})=\Sigma _{i,j}\alpha _{i}\beta _{j}B_{i}B_{j}\,$.

Grade projection

Using a canonical basis, a graded vector space structure can be established. Elements of the geometric algebra that are simply scalar multiples of 1 are grade-0 blades and are called scalars. Nonzero multivectors that are in the span of $\{e_{1},\cdots ,e_{n}\}$ are grade-1 blades and are the ordinary vectors. Multivectors in the span of $\{e_{i}e_{j}\mid 1\leq i<j\leq n\}$ are grade-2 blades and are the bivectors. This terminology continues through to the last grade of n-vectors. Alternatively, grade-n blades are called pseudoscalars, grade-n−1 blades pseudovectors, etc. Many of the elements of the algebra are not graded by this scheme since they are sums of elements of differing grade. Such elements are said to be of mixed grade. The grading of multivectors is independent of the orthogonal basis chosen originally.

A multivector $A$ may be decomposed with the grade-projection operator $\langle A\rangle _{r}$, which outputs the grade-r portion of A. As a result:

$A=\sum _{r=0}^{n}\langle A\rangle _{r}$

As an example, the geometric product of two vectors $ab=a\cdot b+a\wedge b=\langle ab\rangle _{0}+\langle ab\rangle _{2}$ since $\langle ab\rangle _{0}=a\cdot b\,$ and $\langle ab\rangle _{2}=a\wedge b\,$ and $\langle ab\rangle _{i}=0\,$ for i other than 0 and 2.

The decomposition of a multivector $A$ may also be split into those components that are even and those that are odd:

$A^{+}=\langle A\rangle _{0}+\langle A\rangle _{2}+\langle A\rangle _{4}+\cdots$

$A^{-}=\langle A\rangle _{1}+\langle A\rangle _{3}+\langle A\rangle _{5}+\cdots$

This makes the algebra a Z₂-graded algebra or superalgebra with the geometric product. Since the geometric product of two even multivectors is an even multivector, they define an even subalgebra. The even subalgebra of an n-dimensional geometric algebra is isomorphic to a full geometric algebra of (n−1) dimensions. Examples include ${\mathcal {G}}^{+}(2,0)\cong {\mathcal {G}}(0,1)$ and ${\mathcal {G}}^{+}(1,3)\cong {\mathcal {G}}(3,0)$.

Representation of subspaces

Geometric algebra represents subspaces of V as multivectors, and so they coexist in the same algebra with vectors from V. A k-dimensional subspace W of V is represented by taking an orthogonal basis $\{b_{1},b_{2},\cdots b_{k}\}$ and using the geometric product to form the blade D = b₁b₂⋅⋅⋅b_k. There are multiple blades representing W; all those representing W are scalar multiples of D. These blades can be separated into two sets: positive multiples of D and negative multiples of D. The positive multiples of D are said to have the same orientation as D, and the negative multiples the opposite orientation.

Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the outer product that (the restricted class of) n-blades provide but that (the generalized class of) grade-n multivectors do not when n ≥ 4.