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Local graph form at a nonsingular derivative minor

Statement

Let $1 \leq q \leq n$ , let $U \subseteq R^{n}$ be open, and let $F : U \to R^{q}$ be $C^{1}$ in the sense of Continuously differentiable maps, local inverses, and local diffeomorphisms. Suppose $F (x_{*}) = 0$ and some $q \times q$ minor of the derivative matrix $D F (x_{*})$ is invertible. This nonsingular-minor hypothesis is the explicit full-row-rank condition used here. If $q < n$ , then after permuting coordinates $x = (u, v) \in R^{n - q} \times R^{q}$ , there are open neighborhoods $P$ of $u_{*}$ and $Q$ of $v_{*}$ and a unique $C^{1}$ map $φ : P \to Q$ such that

$F (u, v) = 0 ⟺ v = φ (u) ((u, v) \in P \times Q) .$

Its derivative is

$D φ (u) = - D_{v} F (u, φ (u))^{- 1} D_{u} F (u, φ (u)) .$

Thus the local zero set is a graph over $n - q$ free coordinates. If $q = n$ , the zero is locally isolated, so the local graph dimension is zero. Here “local graph dimension $k$ ” means a local graph parameterized by an open subset of $R^{k}$ , with a singleton counted as dimension zero.

Facts & Assumptions

Given: The open domain, $C^{1}$ map, zero, dimensions, and invertible derivative minor in the Statement.

[F1]

Continuously differentiable maps, local inverses, and local diffeomorphisms: a map is $C^{1}$ when it is totally differentiable everywhere and the entries of its derivative matrix are continuous.

[F2]

Invertible Euclidean linear maps: a square linear map is invertible when it has a two-sided linear inverse.

[L1]

The Euclidean implicit function theorem with derivative formula: for an open set $W \subseteq R^{m + q}$ and a $C^{1}$ map $G : W \to R^{q}$ with $m, q \geq 1$ , $(a, b) \in W$ , $G (a, b) = 0$ , and invertible second-block derivative $D_{y} G (a, b)$ , there are open neighborhoods $P, Q$ and a unique $C^{1}$ map $φ : P \to Q$ with $G (u, v) = 0$ exactly when $v = φ (u)$ on $P \times Q$ . After shrinking these neighborhoods, $D_{y} G (u, φ (u))$ is invertible and $D φ (u) = - D_{y} G (u, φ (u))^{- 1} D_{x} G (u, φ (u))$ .

[L2]

The Euclidean inverse function theorem: a $C^{1}$ map between open subsets of $R^{n}$ with invertible derivative at a point is a local $C^{1}$ bijection with a $C^{1}$ inverse.

[L3]

Every Euclidean linear map has a unique matrix and satisfies $∥ L h ∥_{2} \leq K ∥ h ∥_{2}$ for some $K \geq 0$ : linear maps between Euclidean spaces have coordinate matrices and are bounded, hence continuous; a coordinate permutation is an invertible linear map.

[L4]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : composing a $C^{1}$ map with a linear coordinate permutation preserves total differentiability and the derivative formula; [L3] and The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement make that permutation a homeomorphism, so it preserves open domains.

Proof

technique · reduction

1.1F1F2L1L3L4givenconstruct

First suppose $q < n$ . Choose the coordinate permutation that puts the columns of the stated invertible minor into the second block $v \in R^{q}$ , and write the remaining coordinates as $u \in R^{n - q}$ . Composing $F$ with the inverse permutation preserves openness and the $C^{1}$ property by [F1], [L3], and [L4]; its second-block derivative at $(u_{*}, v_{*})$ is exactly the chosen invertible minor in the sense of [F2]. Since $n - q \geq 1$ and $q \geq 1$ , [L1] applies and gives the neighborhoods, unique graph $φ$ , equivalence, and displayed derivative formula.

2.1step 1.1algebra

The graph map $u \mapsto (u, φ (u))$ is one-to-one because its first coordinate is $u$ , and its inverse on the graph is coordinate projection. Its parameter domain $P$ is open in $R^{n - q}$ , so this is the claimed local graph dimension.

3.1F2L2givenalgebra∎

Now suppose $q = n$ . The selected minor is the whole square derivative matrix $D F (x_{*})$ , so it is invertible by [F2]. Apply [L2] on neighborhoods $V$ of $x_{*}$ and $W$ of $0$ . Injectivity of $F ∣_{V}$ and $F (x_{*}) = 0$ imply $F (x) = 0$ for $x \in V$ exactly when $x = x_{*}$ . Thus the local zero set is a singleton and has the stated zero-dimensional graph form.

Local graph form at a nonsingular derivative minor

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources