[quadratic-form] Quadratic form alias [metric] Metric alias Inner product [inner-product]
Example Distance
Example spacetime metric
Bilinear function, Quadratic form
The commonly used notation for metric is . In coordinates, it is denoted as
Under coordinates, metric can be represented as matrix and matrix multiplication
[signature-inertia] signature is invariant under . The eigenvalues and diagonalization of symmetric matrix are where there are ones, zeros, and negative ones. This can be understood as orbit classification of group action
[quadratic-form-non-degenerated] Non-degenerate := in signature
Degenerate quadratic forms can be restricted to the subspace to become non-degenerate
The following are equivalent
- metric is non-degenerate
-
[quadratic-form-dual] is a bijection alias [musical-isomorphism]
The dual map relative to the metric is denoted as , and the inverse map of the metric dual is denoted as
- The quadratic form matrix is invertible
When a non-degenerate quadratic form is fixed, the structure group . Keeping two directions yields
The inverse of the metric matrix is . In coordinates, it is denoted as and
==> In coordinates
let the dual basis of the basis of the vector space be defined as
let then
[rasing-and-lowring-index] Raising and lowering indices
quadratic-form-dual Matrix representation in coordinates
where is the lowered index. Or
Note that the metric matrix is symmetric or
are the coefficients of represented by the dual basis, because , or using are the coefficients of in the dual basis
For the inverse of the metric matrix, define the metric of the dual space , satisfying
In coordinates
The metric dual of is the inverse of the metric dual of
Therefore, there is also a raised index
Conversely, starting from the inverse of , it can be proven that it is the metric dual of some metric in , and this metric is in matrix representation
The metric dual of the metric under the metric of is . The reverse is also true
[tensor] Multilinearity (compatible with and logic of Cartesian product) + minimally independent (generating basis)
Derived basis and coefficients of derived basis
From the properties of tensors
[tensor-induced-quadratic-form]
derive the quadratic form of the vector space to the quadratic form of the tensor space
Traverse all orthogonal bases with
get signature
[rasing-and-lowring-index-tensor] Tensors can also metric dual raise/lower indices
Example Lowering index