---- > [!definition] Definition. ([[musical isomorphism induced by a nondegenerate bilinear form]]) > Let $V$ be a finite-dimensional [[vector space]] endowed with a [[symmetric multilinear map|symmetric]] [[bilinear map|bilinear form]] $\langle -,- \rangle$. With $V^{*}$ denoting the [[dual vector space|dual]] of $V$, define $\begin{align} \flat:V &\to V^{*} \\ v & \mapsto \langle v, - \rangle . \end{align}$ This map is a [[linear isomorphism]] if and only if the form is [[nondegenerate bilinear form|nondegenerate]], if and only if its [[matrix of a bilinear form|matrix]] is [[inverse matrix|invertible]]. The inverse, $\sharp: V^{*} \to V,$ maps $\theta \in V^{*}$ to the unique element $\theta^{\sharp}\in V$ satisfying $\langle \theta^{\sharp}, w \rangle= \theta(w)$ for all $w \in V$. We call $\flat$ and $\sharp$ the **musical isomorphisms** between $V$ and $V^{*}$. > In coordinates: supposing $\langle -,- \rangle$ has matrix $\boldsymbol G=[g_{ij}]$ wrt [[basis]] $(e_{i})$ on $V$ and its [[dual basis|dual]] $(e^{i})$ on $V^{*}$, with inverse $\boldsymbol G^{-1}=[g^{ij}]$, $\sharp$ is given in coordinates by $\boldsymbol \theta^{\sharp}= \theta_{j} \boldsymbol g ^{:, j}, \theta^{\sharp^{i}}=g^{ij}\theta_{j}$ for $\boldsymbol \theta=\theta_{j}e^{j} \in V^{*}$. > [!definition] Between [[vector bundle|bundles]] (differential geometry). > For a [[smooth manifold]] $M$ with [[metric tensor]] $\langle -,- \rangle \in \operatorname{Sym}^{2}(T^{*}M)$, **musical isomorphism** refers to a canonical identification of the [[tangent bundle]] $TM$ and [[cotangent bundle]] $T^{*}M$ as a 'global' version of the discussion above. > > $TM \xrightarrow[\cong]{\flat} T^{*}M.$ > > [!justification] > To check that $\flat$ is an [[isomorphism]], it suffices to observe that the nondegeneracy condition is precisely saying that $\ker \flat=\{ 0 \}$. Hence the map is [[injection|injective]] iff the form is nondegenerate, and since $\dim V = \dim V^{*}$ [[injectivity is equivalent to surjectivity in finite dimensions|it is also surjective]] in this case. As for the matrix claim, it follows just because the matrix of $\flat$ is precisely the matrix of the bilinear form, and hence inherits its properties. ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```