Difference between revisions of "Poly-vector"
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− | + | '' $ p $- | |
+ | vector, over a vector space $ V $'' | ||
− | + | An element of the $ p $- | |
+ | th exterior degree $ \wedge ^ {p} V $ | ||
+ | of the space $ V $ | ||
+ | over a field $ K $( | ||
+ | see [[Exterior algebra|Exterior algebra]]). A $ p $- | ||
+ | vector can be understood as a $ p $- | ||
+ | times skew-symmetrized contravariant tensor on $ V $. | ||
+ | Any linearly independent system of vectors $ x _ {1} \dots x _ {p} $ | ||
+ | from $ V $ | ||
+ | defines a non-zero $ p $- | ||
+ | vector $ x _ {1} \wedge \dots \wedge x _ {p} $; | ||
+ | such a poly-vector is called factorable, decomposable, pure, or prime (often simply a poly-vector). Here two linearly independent systems $ x _ {1} \dots x _ {p} $ | ||
+ | and $ y _ {1} \dots y _ {p} $ | ||
+ | generate the same subspace in $ V $ | ||
+ | if and only if $ y _ {1} \wedge \dots \wedge y _ {p} = cx _ {1} \wedge \dots \wedge x _ {p} $, | ||
+ | where $ c \in K $. | ||
+ | For any non-zero poly-vector $ t \in \wedge ^ {p} V $, | ||
+ | its annihilator $ \mathop{\rm Ann} t = \{ {v \in V } : {t \wedge v = 0 } \} $ | ||
+ | is a subspace of dimension $ \leq p $, | ||
+ | and the poly-vector $ t $ | ||
+ | is pure if and only if $ \mathop{\rm dim} \mathop{\rm Ann} t = p $. | ||
+ | The pure $ p $- | ||
+ | vectors of an $ n $- | ||
+ | dimensional space $ V $ | ||
+ | form an algebraic variety in $ \wedge ^ {p} V $; | ||
+ | the corresponding projective algebraic variety is a [[Grassmann manifold|Grassmann manifold]]. Any non-zero $ n $- | ||
+ | vector or $ ( n- 1) $- | ||
+ | vector in an $ n $- | ||
+ | dimensional space $ V $ | ||
+ | is pure, but a bivector $ t $ | ||
+ | is pure if and only if $ t \wedge t = 0 $. | ||
− | If | + | If $ v _ {1} \dots v _ {n} $ |
+ | is a basis of $ V $ | ||
+ | and $ x _ {i} = \sum_{j=1} ^ {n} x _ {i} ^ {j} v _ {j} $, | ||
+ | then the coordinates of the poly-vector $ t = x _ {1} \wedge \dots \wedge x _ {p} $ | ||
+ | in the basis $ \{ {v _ {i _ {1} } \wedge \dots \wedge v _ {i _ {p} } } : {i _ {1} < \dots < i _ {p} } \} $ | ||
+ | of the space $ \wedge ^ {p} V $ | ||
+ | are the minors $ t ^ {i _ {1} \dots i _ {p} } = \mathop{\rm det} \| x _ {i} ^ {i _ {k} } \| $, | ||
+ | $ i _ {1} < \dots < i _ {p} $, | ||
+ | of the matrix $ \| x _ {i} ^ {j} \| $. | ||
+ | In particular, for $ p = n $, | ||
− | + | $$ | |
+ | x _ {1} \wedge \dots \wedge x _ {n} = \ | ||
+ | \mathop{\rm det} \| x _ {i} ^ {j} \| v _ {1} \wedge \dots \wedge v _ {n} . | ||
+ | $$ | ||
− | + | If one specifies a non-zero $ n $- | |
+ | vector $ \omega \in \wedge ^ {n} V $, | ||
+ | a duality between $ p $- | ||
+ | vectors and $ ( n- p) $- | ||
+ | vectors is obtained, i.e. a natural isomorphism | ||
− | + | $$ | |
+ | \pi : \wedge ^ {p} ( V) \rightarrow \ | ||
+ | ( \wedge ^ {n-p} V) ^ {*} \cong \wedge ^ {n-p} ( V ^ {*} ) | ||
+ | $$ | ||
− | + | such that $ t \wedge u = \pi ( t)( u) \omega $ | |
+ | for all $ t \in \wedge ^ {p} V $ | ||
+ | and $ u \in \wedge ^ {n-p} V $. | ||
− | of a pure poly-vector | + | Let $ k = \mathbf R $ |
+ | and let an inner product be defined in $ V $, | ||
+ | then in $ \wedge ^ {p} V $ | ||
+ | an inner product is induced with the following property: For any orthonormal basis $ v _ {1} \dots v _ {n} $ | ||
+ | in $ V $ | ||
+ | the basis $ \{ {v _ {i _ {1} } \wedge \dots \wedge v _ {i _ {p} } } : {i _ {1} < \dots < i _ {p} } \} $ | ||
+ | in $ \wedge ^ {p} V $ | ||
+ | is also orthonormal. The scalar square | ||
+ | |||
+ | $$ | ||
+ | ( t, t) = \sum _ {i _ {1} < \dots < i _ {p} } | ||
+ | ( t ^ {i _ {1} \dots i _ {p} } ) ^ {2} | ||
+ | $$ | ||
+ | |||
+ | of a pure poly-vector $ t = x _ {1} \wedge \dots \wedge x _ {p} $ | ||
+ | coincides with the square of the volume of the parallelopipedon in $ V $ | ||
+ | constructed on the vectors $ x _ {1} \dots x _ {p} $. | ||
+ | If one specifies an orientation in the $ n $- | ||
+ | dimensional Euclidean space $ V $( | ||
+ | which is equivalent to choosing an $ n $- | ||
+ | vector $ \omega $ | ||
+ | for which $ ( \omega , \omega ) = 1 $), | ||
+ | then the above duality leads to a natural isomorphism $ \gamma : \wedge ^ {p} V \rightarrow \wedge ^ {n-p} V $. | ||
+ | In particular, the $ ( n- 1) $- | ||
+ | vector $ t = x _ {1} \wedge \dots \wedge x _ {n-1} $ | ||
+ | corresponds to a vector $ \gamma ( t) \in V $, | ||
+ | called the vector product of the vectors $ x _ {1} \dots x _ {n-1} $. | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian)</TD></TR> | ||
+ | </table> |
Latest revision as of 18:59, 9 January 2024
$ p $-
vector, over a vector space $ V $
An element of the $ p $- th exterior degree $ \wedge ^ {p} V $ of the space $ V $ over a field $ K $( see Exterior algebra). A $ p $- vector can be understood as a $ p $- times skew-symmetrized contravariant tensor on $ V $. Any linearly independent system of vectors $ x _ {1} \dots x _ {p} $ from $ V $ defines a non-zero $ p $- vector $ x _ {1} \wedge \dots \wedge x _ {p} $; such a poly-vector is called factorable, decomposable, pure, or prime (often simply a poly-vector). Here two linearly independent systems $ x _ {1} \dots x _ {p} $ and $ y _ {1} \dots y _ {p} $ generate the same subspace in $ V $ if and only if $ y _ {1} \wedge \dots \wedge y _ {p} = cx _ {1} \wedge \dots \wedge x _ {p} $, where $ c \in K $. For any non-zero poly-vector $ t \in \wedge ^ {p} V $, its annihilator $ \mathop{\rm Ann} t = \{ {v \in V } : {t \wedge v = 0 } \} $ is a subspace of dimension $ \leq p $, and the poly-vector $ t $ is pure if and only if $ \mathop{\rm dim} \mathop{\rm Ann} t = p $. The pure $ p $- vectors of an $ n $- dimensional space $ V $ form an algebraic variety in $ \wedge ^ {p} V $; the corresponding projective algebraic variety is a Grassmann manifold. Any non-zero $ n $- vector or $ ( n- 1) $- vector in an $ n $- dimensional space $ V $ is pure, but a bivector $ t $ is pure if and only if $ t \wedge t = 0 $.
If $ v _ {1} \dots v _ {n} $ is a basis of $ V $ and $ x _ {i} = \sum_{j=1} ^ {n} x _ {i} ^ {j} v _ {j} $, then the coordinates of the poly-vector $ t = x _ {1} \wedge \dots \wedge x _ {p} $ in the basis $ \{ {v _ {i _ {1} } \wedge \dots \wedge v _ {i _ {p} } } : {i _ {1} < \dots < i _ {p} } \} $ of the space $ \wedge ^ {p} V $ are the minors $ t ^ {i _ {1} \dots i _ {p} } = \mathop{\rm det} \| x _ {i} ^ {i _ {k} } \| $, $ i _ {1} < \dots < i _ {p} $, of the matrix $ \| x _ {i} ^ {j} \| $. In particular, for $ p = n $,
$$ x _ {1} \wedge \dots \wedge x _ {n} = \ \mathop{\rm det} \| x _ {i} ^ {j} \| v _ {1} \wedge \dots \wedge v _ {n} . $$
If one specifies a non-zero $ n $- vector $ \omega \in \wedge ^ {n} V $, a duality between $ p $- vectors and $ ( n- p) $- vectors is obtained, i.e. a natural isomorphism
$$ \pi : \wedge ^ {p} ( V) \rightarrow \ ( \wedge ^ {n-p} V) ^ {*} \cong \wedge ^ {n-p} ( V ^ {*} ) $$
such that $ t \wedge u = \pi ( t)( u) \omega $ for all $ t \in \wedge ^ {p} V $ and $ u \in \wedge ^ {n-p} V $.
Let $ k = \mathbf R $ and let an inner product be defined in $ V $, then in $ \wedge ^ {p} V $ an inner product is induced with the following property: For any orthonormal basis $ v _ {1} \dots v _ {n} $ in $ V $ the basis $ \{ {v _ {i _ {1} } \wedge \dots \wedge v _ {i _ {p} } } : {i _ {1} < \dots < i _ {p} } \} $ in $ \wedge ^ {p} V $ is also orthonormal. The scalar square
$$ ( t, t) = \sum _ {i _ {1} < \dots < i _ {p} } ( t ^ {i _ {1} \dots i _ {p} } ) ^ {2} $$
of a pure poly-vector $ t = x _ {1} \wedge \dots \wedge x _ {p} $ coincides with the square of the volume of the parallelopipedon in $ V $ constructed on the vectors $ x _ {1} \dots x _ {p} $. If one specifies an orientation in the $ n $- dimensional Euclidean space $ V $( which is equivalent to choosing an $ n $- vector $ \omega $ for which $ ( \omega , \omega ) = 1 $), then the above duality leads to a natural isomorphism $ \gamma : \wedge ^ {p} V \rightarrow \wedge ^ {n-p} V $. In particular, the $ ( n- 1) $- vector $ t = x _ {1} \wedge \dots \wedge x _ {n-1} $ corresponds to a vector $ \gamma ( t) \in V $, called the vector product of the vectors $ x _ {1} \dots x _ {n-1} $.
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) |
[2] | A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian) |
[3] | M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian) |
Poly-vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-vector&oldid=17085