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The simple-fold complement restriction is an isomorphism

Statement

For the data and notation of Complement splitting and range projection for a simple fold, $J ∣_{X} : X \to Y$ is a linear isomorphism. The map $P = I - q w^{T}$ satisfies $P^{2} = P, range P = Y, \ker P = span {q}, P J = J, P b = 0.$

Facts & Assumptions

Given: $J, v, w, b, a, X, Y, q, P$ and their hypotheses in Complement splitting and range projection for a simple fold.

[F1]

Rank-nullity: $\dim_{F} V = nullity T + rank T$ : for a map between finite-dimensional spaces, $\dim V = \dim \ker T + \dim im T$ .

[F2]

In finite dimension, $W^{⊥ ⊥} = W$ and $\dim W + \dim W^{⊥} = \dim V$ : $\dim W + \dim W^{⊥} = \dim V$ in finite-dimensional inner-product spaces.

[F3]

If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ : subspaces of equal finite dimension are equal; The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial: a linear map is injective exactly when its kernel is zero.

[F4]

The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ : $\dim R^{n} = n$ .

[F5]

Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, and Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis: independence means a zero linear combination has only zero coefficients, a basis is an independent spanning set, and dimension is the size of a finite basis.

[F6]

Rank and nullity of a linear map with finite-dimensional domain: rank of a map with finite-dimensional domain is the dimension of its image.

Proof

Given: The fold data in Complement splitting and range projection for a simple fold.

1.1F2F5givenalgebra

Since $v$ is unit it is nonzero. If $α v = 0$ , then $α ∥ v ∥_{2}^{2} = 0$ , so $α = 0$ ; thus ${v}$ is independent, and it spans $span {v}$ by definition. It is therefore a basis and that span has dimension one by [F5]. The orthogonal-complement formula [F2] gives $\dim X = n - 1$ .

1.2F1F4givenalgebra

Since $w^{T} q = 1$ , the map $w^{T}$ is onto $R$ . Rank-nullity [F1] and $\dim R^{n} = n$ [F4] give $\dim Y = n - 1$ ; [F4] also gives $\dim R^{n} = n$ .

1.3givenalgebra

Since $w^{T} q = 1$ , direct multiplication gives $P^{2} = (I - q w^{T})^{2} = I - q w^{T} = P$ . Also $w^{T} P = 0$ , so $range P \subseteq Y$ , while $P y = y$ for every $y \in Y$ ; hence $range P = Y$ . The equation $P z = 0$ is equivalent to $z = q (w^{T} z)$ , so $\ker P = span {q}$ .

2.1F1F3F6step 1.1step 1.2givenalgebra

From $w^{T} J = 0$ we have $im J \subseteq Y$ , and $J$ has rank $n - 1$ by [F6]. Thus $\dim im J = \dim Y$ , so [F3] gives $im J = Y$ . Moreover, if $x \in X \cap \ker J$ , then $x = α v$ and $0 = v^{T} x = α$ , so $J ∣_{X}$ is injective by [F3]. Its image has dimension $n - 1$ by [F1] and therefore equals $Y$ by [F3]. This proves that $J ∣_{X}$ is an isomorphism.

3.1givenstep 1.3algebra∎

Finally $P J = J - q (w^{T} J) = J$ , and $P b = b - q (w^{T} b) = b - q a = 0$ by the definitions.

Depends on

Used by

A simple fold has a unique local Lyapunov–Schmidt chart and nonzero parameter curvature Theorem

Dependency tree · two levels

49 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eugene L. Allgower and Kurt Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics 13 (1990) (standard reference, not scraped)
K. A. Cliffe, A. Spence, and S. J. Tavener, The numerical analysis of bifurcation problems with application to fluid mechanics, Acta Numerica 9 (2000), 39–131 (standard reference, not scraped)