Editing Geometric algebra (section)

=== Blades, grades, and basis ===
A multivector that is the exterior product of <math>r</math> linearly independent vectors is called a ''blade'', and is said to be of grade {{tmath|1= r }}.{{efn|Grade is a synonym for ''degree'' of a homogeneous element under the [[graded algebra|grading as an algebra]] with the exterior product (a {{tmath|1= \mathrm{Z} }}-grading), and not under the geometric product.}} A multivector that is the sum of blades of grade <math>r</math> is called a (homogeneous) multivector of grade {{tmath|1= r }}.  From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.

Consider a set of <math>r</math> linearly independent vectors <math>\{a_1,\ldots,a_r\}</math> spanning an {{tmath|1= r }}-dimensional subspace of the vector space. With these, we can define a real [[symmetric matrix]] (in the same way as a [[Gramian matrix]])
: <math>[\mathbf{A}]_{ij} = a_i \cdot a_j</math>

By the [[spectral theorem]], <math>\mathbf{A}</math> can be diagonalized to [[diagonal matrix]] <math>\mathbf{D}</math> by an [[orthogonal matrix]] <math>\mathbf{O}</math> via
: <math>\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}</math>

Define a new set of vectors {{tmath|1= \{e_1, \ldots,e_r\} }}, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:
: <math>e_i=\sum_j[\mathbf{O}]_{ij}a_j</math>

Since orthogonal transformations preserve inner products, it follows that <math>e_i\cdot e_j=[\mathbf{D}]_{ij}</math> and thus the <math>\{e_1, \ldots, e_r\}</math> are perpendicular. In other words, the geometric product of two distinct vectors <math>e_i \ne e_j</math> is completely specified by their exterior product, or more generally
: <math>\begin{array}{rl}
e_1e_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}</math>

Therefore, every blade of grade <math>r</math> can be written as the exterior product of <math>r</math> 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 [[unit vector|normalized]] according to
: <math>\widehat{e_i}=\frac{1}{\sqrt{|e_i \cdot e_i|}}e_i,</math>
then these normalized vectors must square to <math>+1</math> or {{tmath|1= -1 }}. By [[Sylvester's law of inertia]], the total number of {{tmath|1= +1 }} and the total number of {{tmath|1= -1 }}s along the diagonal matrix is invariant. By extension, the total number <math>p</math> of these vectors that square to <math>+1</math> and the total number <math>q</math> that square to <math>-1</math> is invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra {{tmath|1= \mathcal{G}(p,q) }}. For example, <math>\mathcal{G}(3,0)</math> models three-dimensional [[Euclidean space]], <math>\mathcal{G}(1,3)</math> relativistic [[spacetime]] and <math>\mathcal{G}(4,1)</math> a [[conformal geometric algebra]] of a three-dimensional space.

The set of all possible products of <math>n</math> orthogonal basis vectors with indices in increasing order, including <math>1</math> as the empty product, forms a basis for the entire geometric algebra (an analogue of the [[Poincaré–Birkhoff–Witt theorem|PBW theorem]]). For example, the following is a basis for the geometric algebra {{tmath|1= \mathcal{G}(3,0) }}:
: <math>\{1, e_1, e_2, e_3, e_1e_2, e_2e_3, e_3e_1, e_1e_2e_3\}</math>
A basis formed this way is called a '''standard basis''' for the geometric algebra, and any other orthogonal basis for <math>V</math> will produce another standard basis. Each standard basis consists of <math>2^n</math> elements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements. If the standard basis elements are <math>\{ B_i \mid i \in S \}</math> with <math>S</math> being an index set, then the geometric product of any two multivectors is
: <math> \left( \sum_i \alpha_i B_i \right) \left( \sum_j \beta_j B_j \right) = \sum_{i,j} \alpha_i\beta_j B_i B_j .</math>

The terminology "<math>k</math>-vector" is often encountered to describe multivectors containing elements of only one grade. In higher dimensional space, some such multivectors are not blades (cannot be factored into the exterior product of <math>k</math> vectors). By way of example, <math> e_1 \wedge e_2 + e_3 \wedge e_4 </math> in <math>\mathcal{G}(4,0)</math> cannot be factored; typically, however, such elements of the algebra do not yield to geometric interpretation as objects, although they may represent geometric quantities such as rotations. Only {{tmath|1= 0 }}-, {{tmath|1= 1 }}-, {{tmath|1= (n-1) }}- and {{tmath|1= n }}-vectors are always blades in {{tmath|1= n }}-space.