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A simple eigenpair has a regularity-matched gauge-fixed parameter branch

Statement

Let $k \geq 1$ and let $A (λ) \in C^{d \times d}$ be a $C^{k}$ matrix path. Let $ζ_{0}$ be a simple eigenvalue of $A (0)$ , and choose compatible left and right eigenvectors $p_{0}, q_{0}$ with $p_{0}^{*} q_{0} = 1$ . Then there are unique local $C^{k}$ functions $ζ (λ), q (λ)$ such that $A (λ) q (λ) = ζ (λ) q (λ), p_{0}^{*} q (λ) = 1, ζ (0) = ζ_{0}, q (0) = q_{0} .$ There is also a unique local $C^{k}$ left eigenvector $p (λ)$ with $A (λ)^{*} p (λ) = \overline{ζ (λ)} p (λ), p (λ)^{*} q (λ) = 1, p (0) = p_{0} .$ Moreover, $ζ^{'} (0) = p_{0}^{*} A^{'} (0) q_{0}$ .

Facts & Assumptions

Given: $A$ , $ζ_{0}$ , $p_{0}$ , $q_{0}$ and the hypotheses in the Statement.

[F1]

Compatible left and right eigenvectors for a simple eigenvalue: $A (0) q_{0} = ζ_{0} q_{0}$ and $p_{0}^{*} A (0) = ζ_{0} p_{0}^{*}$ .

[F2]

For a simple eigenvalue, left and right eigenvectors pair nontrivially and may be normalized by $y^{*} x = 1$ : compatible left and right eigenvectors for a simple eigenvalue can be normalized by $p_{0}^{*} q_{0} = 1$ .

[F3]

The parametrized implicit function theorem with $C^{k}$ regularity: a $C^{k}$ equation with an invertible derivative in the solved variables has a unique local $C^{k}$ solution branch for every $k \geq 1$ .

[F4]

A simple eigenvalue and a gauge-fixed right eigenvector admit local $C^{1}$ branches in the underlying real matrix space: the simple eigenvalue and the same gauge-fixed eigenvector have unique local $C^{1}$ branches.

[F5]

Along a differentiable matrix path, a simple eigenvalue satisfies $λ^{'} = y^{*} A^{'} x$ under the normalization $y^{*} x = 1$ : along compatible branches normalized by $p^{*} q = 1$ , $ζ^{'} = p^{*} A^{'} q$ .

Proof

Given: The data and hypotheses in the Statement.

1.1F1F2givenalgebra

Regard complex vector spaces as real spaces of twice the dimension and define $Φ (λ, ζ, q) : = ((A (λ) - ζ I) q, p_{0}^{*} q - 1) .$ This is $C^{k}$ by the matrix path hypothesis. Its derivative in $(ζ, q)$ at $(0, ζ_{0}, q_{0})$ sends $(a, v)$ to $((A (0) - ζ_{0} I) v - a q_{0}, p_{0}^{*} v)$ . If this derivative is zero, left multiplication of the first component by $p_{0}^{*}$ gives $- a = 0$ by [F1] and [F2]; then $v$ is in the eigenspace spanned by $q_{0}$ , and $p_{0}^{*} v = 0$ forces $v = 0$ . Thus the derivative is injective and, its finite-dimensional domain and codomain having equal real dimension, invertible.

2.1F3F4step 1.1

Apply [F3] to $Φ = 0$ in the solved variables $(ζ, q)$ . It gives unique $C^{k}$ branches satisfying the eigenvalue equation, gauge, and base values. By uniqueness they agree with the $C^{1}$ branches in [F4].

2.2F1F2F3step 1.1algebra

Apply the same implicit-function construction as step 1.1 to $A (λ)^{*}$ at eigenvalue ${\overset{ˉ}{ζ}}_{0}$ , with gauge $q_{0}^{*} p (λ) = 1$ . The same injectivity computation, with $q_{0}, p_{0}$ interchanged and conjugated, gives a unique $C^{k}$ left eigenvector branch $\tilde{p} (λ)$ ; its eigenvalue is $\overline{ζ (λ)}$ by uniqueness of the simple local branch.

3.1F1F2step 2.1step 2.2givenalgebra

Set $c (λ) = \tilde{p} (λ)^{*} q (λ)$ . It is $C^{k}$ and $c (0) = 1$ , so it stays nonzero after shrinking the interval. Define $p (λ) = \tilde{p} (λ) / \overline{c (λ)}$ ; then $p (λ)^{*} q (λ) = c (λ) / c (λ) = 1$ and its eigenvector equation is unchanged. If $\hat{p}$ is another such local branch, rescale it by $d (λ) = q_{0}^{*} \hat{p} (λ)$ , which is nonzero near zero and equals $1$ at zero. Then $\hat{p} / d$ satisfies the fixed gauge $q_{0}^{*} (\hat{p} / d) = 1$ and solves the same adjoint eigenpair equation, so IFT uniqueness from step 2.2 gives $\hat{p} / d = \tilde{p}$ . The normalization ${\hat{p}}^{*} q = 1$ then forces $d = 1 / \overline{c}$ , hence $\hat{p} = p$ .

4.1F1F2F5step 2.1algebra∎

Differentiate $A (λ) q (λ) = ζ (λ) q (λ)$ at zero and left-multiply by $p_{0}^{*}$ . The terms containing $q^{'} (0)$ cancel by [F1], and $p_{0}^{*} q_{0} = 1$ by [F2], leaving $ζ^{'} (0) = p_{0}^{*} A^{'} (0) q_{0}$ , also the interface stated in [F5].

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Sources

Yuri A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer (standard reference, not scraped)
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, 1976 (standard reference, not scraped)