[quadratic_form] quadratic form alias [metric] metric alias inner product [inner_product]
Example distance
Example spacetime metric
Bilinear function, quadratic form
metric common notations . In coordinates written as
In coordinates metric can be expressed as matrix and matrix multiplication
[signature_inertia] signature invariant under . Eigenvalues and diagonalization of symmetric matrix with โs, โs, โs.
Different signatures can be understood as classification of orbits of the group action on the space of quadratic forms
[quadratic_form_non_degenerated] non-degenerate := in signature
Degenerate quadratic form can be restricted to subspace to become non-degenerate
The following are equivalent
- metric is non-degenerate
-
[quadratic_form_dual] is a bijection alias [musical_isomorphism]
Dual mapping relative to metric denoted , inverse of metric dual mapping denoted
- The quadratic form matrix is invertible
When fixing a non-degenerate quadratic form, the structure group . Preserving both orientations yields
The inverse of the metric matrix . In coordinates, denoted as and
==> In coordinates
let the dual basis of the basis :=
let then
[rasing_and_lowring_index] Raising and lowering indices
quadratic-form-dual Matrix representation in coordinates
where is lowering an index. Or
Note the metric matrix is symmetric or
are the coefficients of the dual basis representation of , because , or using that are the coefficients of in the dual basis
For the inverse of the metric matrix, define the metric of the dual space as . It satisfies the following equation
The equation is expressed in coordinates as
The metric dual of is the inverse of the metric dual of , and thus is also
Therefore, there is also raising of indices
[tensor] Multilinear (compatible with the and logic of set product) + minimal independent generation (quotient construction)
Derived basis , coefficients of the derived basis
From the properties of tensors
[tensor_induced_quadratic_form]
Extend the quadratic form of the vector space to the quadratic form of the tensor space
Iterate over all orthogonal bases with
Obtain the signature
[rasing_and_lowring_index_tensor] Tensors can also raise and lower indices via metric dual
Example Lowering an index
in , the metric dual of the induced metric is . The converse also holds
Denote as
Denote the quadratic form as
Consider simple tensors of . For , the induced metric on the tensor is
Prop trace-identity duality
Proof
Question Is there a more intrinsic, coordinate-free proof method?
The first proof method is to use an orthonormal basis
Then
It suffices to prove the case for
The second proof uses the result for tensors, then converts back to tensors
For tensors, as the dual of
Prop let , then
Proof Take an orthonormal basis , then
let defined as
Prop
Proof
Prop
Proof It suffices to prove the case for simple tensors
But
Therefore
The last step can be clarified further. For
Prop
Proof
Prop
Proof Apply to both sides
Prop Using matrix to represent
- corresponds to
- corresponds to
- Thus the matrix representation of is
Prop
Prop In an orthonormal basis and . In this case
Prop
Prop (in an orthonormal basis)
Prop
Proof and
Prop [Killing_form_of_orthogonal_group] (Killing form of up to a constant)
Proof According to