note-math

quadratic-form_(tag) $ℝ^{𝑛}$ Quadratic form alias metric_(tag) Metric alias Inner product inner-product_(tag)

Example $ℝ^{2}, ℝ^{3}$ Distance $(\begin{matrix} 𝑥 \\ 𝑦 \\ 𝑧 \end{matrix}) ⇝ 𝑥^{2} + 𝑦^{2} + 𝑧^{2}$

Example $ℝ^{1, 3}$ spacetime metric $(\begin{matrix} 𝑡 \\ 𝑥 \\ 𝑦 \\ 𝑧 \end{matrix}) ⇝ 𝑐^{2} \cdot 𝑡^{2} - (𝑥^{2} + 𝑦^{2} + 𝑧^{2})$

Bilinear function, Quadratic form

The commonly used notation for metric is $𝑔 (𝑥, 𝑦), ⟨ 𝑥, 𝑦 ⟩$ . In coordinates, it is denoted as $𝑔_{𝑖 𝑗} 𝑥_{𝑖} 𝑦_{𝑗}$

Under coordinates, metric $𝑔$ can be represented as matrix $𝐺$ and matrix multiplication $(𝑋, 𝑌) ⇝ 𝑋^{⊺} 𝐺 𝑌$

signature-inertial_(tag) signature $(𝑝, 𝑞)$ is invariant under $GL (𝑛, ℝ)$ . The eigenvalues and diagonalization of symmetric matrix $𝐺$ are $(\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix})$ where there are $𝑛_{+}$ ones, $𝑛_{0}$ zeros, and $𝑛_{-}$ negative ones. This can be understood as orbit classification of group action

quadratic-form-non-degenerated_(tag) Non-degenerate := $𝑁_{0} = 0$ in signature

Degenerate quadratic forms can be restricted to the $𝑛_{+} + 𝑛_{-}$ subspace to become non-degenerate

The following are equivalent

metric is non-degenerate
quadratic-form-dual_(tag) is a bijection
$\begin{matrix} 𝑉 & ⟶ & Lin (𝑉 \to ℝ) or 𝑉^{⊺} \\ 𝑣 & ⟿ & ⟨ 𝑣, ⟩ or ⟨ 𝑣 | or 𝑔 (𝑣,) or 𝑣^{♭} \end{matrix}$
The dual map relative to the metric is denoted as $𝑣^{♭}$ , and the inverse map of the metric dual is denoted as $𝛼^{♯}$
The quadratic form matrix is invertible
$det 𝐺 \neq 0$

When a non-degenerate quadratic form is fixed, the structure group $≃ O (𝑝, 𝑞)$ . Keeping two directions yields $SO (𝑝, 𝑞)$

The inverse of the metric matrix $𝑔$ is $𝑔^{- 1}$ . In coordinates, it is denoted as $𝑔 ⇝ 𝑔_{𝑖 𝑖^{'}}$ and $𝑔^{- 1} ⇝ 𝑔^{𝑖 𝑖^{'}}$

$𝑔 \cdot 𝑔^{- 1} = 𝟙 = 𝑔^{- 1} \cdot 𝑔$ ==> In coordinates $𝑔_{𝑖 𝑗} 𝑔^{𝑖^{'} 𝑗} = 𝛿_{𝑖}^{𝑖^{'}} = 𝑔^{𝑖^{'} 𝑗} 𝑔_{𝑖 𝑗}$

let the dual basis $𝑒^{1}, \dots, 𝑒^{𝑛} \in (𝑉 \to ℝ)$ of the basis $𝑒_{1}, \dots, 𝑒_{𝑛} \in 𝑉$ of the vector space be defined as $𝑒^{𝑖} (𝑒_{𝑗}) = 𝛿_{𝑗}^{𝑖}$

let $𝛼 = 𝛼_{𝑖} 𝑒^{𝑖}$ then

\begin{matrix} 𝛼 (𝑒_{𝑗}) & = & 𝛼_{𝑖} 𝑒^{𝑖} (𝑒_{𝑗}) \\ = & 𝛼_{𝑖} 𝛿_{𝑗}^{𝑖} \\ = & 𝛼_{𝑗} \end{matrix}

rasing-and-lowring-index_(tag) Raising and lowering indices

quadratic-form-dual Matrix representation in coordinates

\begin{matrix} 𝑔 (𝑣,) ⇝ 𝑣^{⊺} 𝑔 & = & (\begin{matrix} 𝑣^{1} & \dots & 𝑣^{𝑛} \end{matrix}) (\begin{matrix} 𝑔_{11} & \dots & 𝑔_{1 𝑛} \\ ⋮ & ⋱ & ⋮ \\ 𝑔_{𝑛 1} & \dots & 𝑔_{𝑛 𝑛} \end{matrix}) \\ = & (\begin{matrix} 𝑣^{𝑗} 𝑔_{𝑗 1} & \dots & 𝑣^{𝑗} 𝑔_{𝑗 𝑛} \end{matrix}) ≔ (\begin{matrix} 𝑣_{1} & \dots & 𝑣_{𝑛} \end{matrix}) \end{matrix}

where $𝑣^{𝑗} 𝑔_{𝑖 𝑗} = 𝑣_{𝑖}$ is the lowered index. Or

\begin{matrix} 𝑔 (, 𝑣) ⇝ 𝑔 𝑣 & = & (\begin{matrix} 𝑔_{11} & \dots & 𝑔_{1 𝑛} \\ ⋮ & ⋱ & ⋮ \\ 𝑔_{𝑛 1} & \dots & 𝑔_{𝑛 𝑛} \end{matrix}) (\begin{matrix} 𝑣^{1} \\ ⋮ \\ 𝑣^{𝑛} \end{matrix}) \\ = & (\begin{matrix} 𝑣^{𝑗} 𝑔_{1 𝑗} \\ ⋮ \\ 𝑣^{𝑗} 𝑔_{𝑛 𝑗} \end{matrix}) ≔ (\begin{matrix} 𝑣_{1} \\ ⋮ \\ 𝑣_{𝑛} \end{matrix}) \end{matrix}

Note that the metric matrix is symmetric $𝑔^{⊺} = 𝑔$ or $𝑔_{𝑖 𝑗} = 𝑔_{𝑗 𝑖}$

$𝑣_{𝑖}$ are the coefficients of $𝑣^{♭}$ represented by the dual basis, because $𝑔 (𝑣, 𝑒_{𝑖}) = 𝑣^{𝑗} 𝑔_{𝑖 𝑗} = 𝑣_{𝑖}$ , or using $𝑔_{𝑖 𝑗}$ are the coefficients of $𝑔$ in the dual basis $𝑒^{𝑖} \otimes 𝑒^{𝑗}$

For the inverse of the metric matrix, define the metric $𝑔^{- 1}$ of the dual space $𝑉^{⊺}$ , satisfying

\begin{matrix} 𝑔^{- 1} (𝑣^{♭}, 𝑤^{♭}) & = & {(𝑔 𝑣)}^{⊺} 𝑔^{- 1} (𝑔 𝑤) \\ = & 𝑣^{⊺} (𝑔 𝑔^{- 1} 𝑔) 𝑤 \\ = & 𝑣^{⊺} 𝑔 𝑤 \\ = & 𝑔 (𝑣, 𝑤) \end{matrix}

In coordinates

$𝑔^{𝑖 𝑖^{'}} 𝛼_{𝑖} 𝛽_{𝑖^{'}}$
$𝑔_{𝑖 𝑖^{'}} 𝛼^{𝑖} 𝛽^{𝑖^{'}}$
$𝛼_{𝑖} 𝛽^{𝑖}$
$𝛼^{𝑖} 𝛽_{𝑖}$

The metric dual of $𝑔^{- 1}$ is the inverse $♯$ of the metric dual $♭$ of $𝑔$

\begin{matrix} 𝑣 & ⇝ & 𝑔 𝑣 \\ ⇝ & 𝑔^{- 1} 𝑔 𝑣 \\ = & 𝑣 \end{matrix}

Therefore, there is also a raised index $𝑔^{- 1} (𝑣^{♭}, 𝑒_{𝑗}) = 𝑣_{𝑗} 𝑔^{𝑖 𝑗} = 𝑣^{𝑖}$

Conversely, starting from the inverse $♯$ of $♭$ , it can be proven that it is the metric dual of some metric in $𝑉^{⊺}$ , and this metric is $𝑔^{- 1}$ in matrix representation

The metric dual of the metric $𝑔^{- 1} = 𝑔^{𝑖 𝑗} 𝑒_{𝑖} 𝑒_{𝑗}$ under the metric of ${(𝑉^{⊺ ⊙ 2})}^{⊺}$ is $𝑔 = 𝑔_{𝑖 𝑗} 𝑒^{𝑖} 𝑒^{𝑗} \in {(𝑉^{⊙ 2})}^{⊺}$ . The reverse is also true

tensor_(tag) Multilinearity (compatible with and logic of Cartesian product) + minimally independent (generating basis)

\begin{matrix} 𝑉_{1} \times \dots \times 𝑉_{𝑘} & ⟶ & 𝑉_{1} \otimes \dots \otimes 𝑉_{𝑘} \\ (𝑣_{1}, \dots, 𝑣_{𝑘}) & ⟿ & 𝑣_{1} \otimes \dots \otimes 𝑣_{𝑘} \end{matrix}

Derived basis $𝑒_{1, 𝑖_{1}} \otimes \dots \otimes 𝑒_{𝑘, 𝑖_{𝑘}}$ and coefficients of derived basis $𝑇^{𝑖_{1} \dots 𝑖_{𝑘}}$

From the properties of tensors

\begin{matrix} 𝑣_{1} \otimes \dots \otimes 𝑣_{𝑘} & = & (𝑣_{1}^{𝑖_{1}} 𝑒_{1, 𝑖_{1}}) \otimes \dots \otimes (𝑣_{𝑘}^{𝑖_{𝑘}} 𝑒_{𝑘, 𝑖_{𝑘}}) \\ = & (𝑣_{1}^{𝑖_{1}} \dots 𝑣_{𝑘}^{𝑖_{𝑘}}) 𝑒_{1, 𝑖_{1}} \otimes \dots \otimes 𝑒_{𝑘, 𝑖_{𝑘}} \end{matrix}

tensor-induced-quadratic-form_(tag)

derive the quadratic form $𝑔$ of the vector space to the quadratic form of the tensor space $𝑉^{\otimes 𝐼} \otimes 𝑉^{⊺ \otimes 𝐽}$

𝑔 (⨂_{𝑖 = 1 . . 𝐼} 𝑣_{𝑖} \otimes ⨂_{𝑗 = 1 . . 𝑗} 𝛼_{𝑗}, ⨂_{𝑖 = 1 . . 𝐼} 𝑤_{𝑖} \otimes ⨂_{𝑗 = 1 . . 𝑗} 𝛽_{𝑗}) = \prod_{𝑖 = 1 . . 𝐼} 𝑔 (𝑣_{𝑖}, 𝑤_{𝑖}) \prod_{𝑖 = 1 . . 𝐽} 𝑔^{- 1} (𝛼_{𝑗}, 𝛽_{𝑗})

Traverse all orthogonal bases $⨂_{𝑖 = 1 . . 𝐼} 𝑒_{𝑘_{𝑖}} \otimes ⨂_{𝑗 = 1 . . 𝐽} 𝑒^{𝑙_{𝑗}}$ with

{⟨ ⨂_{𝑖 = 1 . . 𝐼} 𝑒_{𝑘_{𝑖}} \otimes ⨂_{𝑗 = 1 . . 𝐽} 𝑒^{𝑙_{𝑗}} ⟩}^{2} = \prod_{𝑖 = 1 . . 𝐼} {⟨ 𝑒_{𝑘_{𝑖}} ⟩}^{2} \prod_{𝑗 = 1 . . 𝐽} {⟨ 𝑒^{𝑙_{𝑗}} ⟩}^{2}

get signature

rasing-and-lowring-index-tensor_(tag) Tensors can also metric dual raise/lower indices

Example Lowering index $𝑇_{𝑗}^{𝑖} ⇝ 𝑔_{𝑖 𝑖^{'}} 𝑇_{𝑗}^{𝑖^{'}} = 𝑇_{𝑖 𝑗}$