The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form: where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,qSee also:

Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

Read more here: » Clifford algebra: Encyclopedia II - Clifford algebra - Examples: Real and complex Clifford algebras

If the dimension of V is n and {e1,…,en} is a basis of V, then the set is a basis for Cℓ(V,Q). The empty product (k = 0) is defined as the multiplicative identity element. For each value of k there are n choose k basis elements, so the total dimension of the Clifford algebra is Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. A ...

See also:

Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

Read more here: » Clifford algebra: Encyclopedia II - Clifford algebra - Basis and dimension

To demonstrate how to determine if a system is time-invariant then consider the two systems: System A: System B: Since system A explicitly depends on t outside of x(t) and y(t) then it is time-variant. System B, however, does not depend explicitly on t so it is time-invariant. ...

See also:

Time-invariant system, Time-invariant system - Simple example, Time-invariant system - Formal example, Time-invariant system - Abstract example

Read more here: » Time-invariant system: Encyclopedia II - Time-invariant system - Simple example

A more formal proof of why system A & B from above is now presented. To perform this proof, the second definition will be used. System A: Start with a delay of the input Now delay the output by δ Clearly , therefore the system is not time-invariant. System B: Start with a delay of the input Now delay the output by ...

See also:

Time-invariant system, Time-invariant system - Simple example, Time-invariant system - Formal example, Time-invariant system - Abstract example

Read more here: » Time-invariant system: Encyclopedia II - Time-invariant system - Formal example

Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q. The Clifford algebra Cℓ(V,Q) is the "freest" algebra generated by V subject to the condition1 If the characteristic of the ground field K is not 2, then one can rewri ...

See also:

Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

Read more here: » Clifford algebra: Encyclopedia II - Clifford algebra - Introduction and basic properties

Let V be a vector space over a field K, and let Q : V → K be a quadratic form on V. In most cases of interest the field K is either R or C (which have characteristic 0) or a finite field. A Clifford algebra Cℓ(V,Q) is a unital associative algebra over K together with a linear map i : V → Cℓ(V,Q) defined by the following universal property: Given any associative algebra A over K a ...

See also:

Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

Read more here: » Clifford algebra: Encyclopedia II - Clifford algebra - Universal property and construction

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold M to M itself. Exponential map - Definition. For v ∈ TpM, there is a unique geodesic γv satisfying γv(0) = p such that the tangent vector γ′v(0) = v. Then the corresponding exponential map is defined by expp(v) = ...

See also:

Exponential map, Exponential map - Lie theory, Exponential map - Definition, Exponential map - Properties, Exponential map - Riemannian geometry, Exponential map - Definition, Exponential map - Properties, Exponential map - Relationships

Read more here: » Exponential map: Encyclopedia II - Exponential map - Riemannian geometry