A theory for optimal regularization in the finite dimensional case

Consider the matrix problem $\text{Ax = y + ε =}\text{y}\text{?}$ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized “ridge” estimates,$\text{x}\text{?}{}_{\mathrm{\lambda }}\text{= (A}{}^1\text{A + λI)}{}^{-1}\text{A}{}^1\text{y}\text{?}$,where ¹ denotes matrix transpose. Of great concern is the determination of the value of λ for which x?_λ “best” approximates $\text{x}{}_{\mathrm{<mtext>0</mtext>}}\text{= A}{}^{\mathrm{+}}\text{y}$. Let $\text{Q = 6}\text{x}\text{?}{}_{\mathrm{\lambda }}\text{? x}{}_{\mathrm{<mtext>0</mtext>}}\text{6}{}^2$,and define λ₀ to be the value of λ for which Q is a minimum. We look for λ₀ among solutions of dQ/dλ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ₀ as a function of y and ε. Theorems involving “noise to signal ratios” determine when λ₀ exists and define the cases λ₀ > 0 and λ₀ = ∞. Estimates for λ₀ and the minimum square error Q₀ = Q(λ₀) are derived.     

Permanental Polytopes of Doubly Stochastic Matrices

A subpolytope Γ of the polytope Ω_n of all n×n nonnegative doubly stochastic matrices is said to be a permanental polytope if the permanent function is constant on Γ. Geometrical properties of permanental polytopes are investigated. No matrix of the form 1⊕A where A is in Ω₂ is contained in any permanental polytope of Ω₃ with positive dimension. There is no 3-dimensional permanental polytope of Ω₃, and there is essentially a unique maximal 2-dimensional permanental polytope of Ω₃ (a square of side $\text{1}\text{3}$). Permanental polytopes of dimension $\text{(n}{}^2\text{?3n+4)}\text{2}$ are exhibited for each n?4.     

Extensions of factorial states of C1-algebras

Let A be a $\text{C}{}^1$-algebra, B be a $\text{C}{}^1$-subalgebra of A, and φ be a factorial state of B. Sometimes, φ may be extended to a factorial state of A by a tensor product method of Sakai (“$\text{C}{}^1$-algebras and $\text{W}{}^1$-algebras, Springer-Verlag, Berlin/Heidelberg/ New York 1971”). Sometimes, there is a weak expectation of A into $\text{π}{}_{\mathrm{\phi }}\text{(B)}$, and then factorial extensions may be found by a method of Sakai and Tsui (Yokohama Math. J.29 (1981), 157–160). These two methods are shown to have the same effect, and the factorial extensions produced by them are analysed.     

Singular sets of Sobolev functionsEnsembles singuliers des fonctions de Sobolev

We are interested in finding Sobolev functions with “large” singular sets. Given $\text{N,k∈}\text{N}$, 1<p<∞, kp<N, for any compact subset A of $\text{R}{}^{\mathrm{N}}$, such that its upper box dimension is less than N?kp, we construct a Sobolev function $\text{u∈}\text{W}{}^{\mathrm{k,p}}\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N?kp. To cite this article: D. ?ubrini?, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.     

On the Minimum Value of the Permanent of a Nearly Decomposable Doubly Stochastic Matrix

Let A be a minimizing matrix for the permanent over the face of Ω_n determined by a fully indecomposable matrix. It is shown that A is fully indecomposable and positive elements of A have permanental minors equal to per(A). Furthermore a zero entry of A has its permanental minor greater than or equal to per(A), provided that same element of the face has its corresponding entry positive. For 2?n?9 the minimum value of the permanent of a nearly decomposable $\text{A∈Ω}{}_{\mathrm{n}}$ is $\text{1}\text{2}{}^{\mathrm{n-1}}$.     

Canonical forms for linear systems

This paper considers canonical forms for the similarity action of Gl(n) on $\text{∑}{}_{\mathrm{n,m}}\text{={(A,B)∈}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{n}}\text{×}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{m}}\text{}}$:

$\text{Gl(n×∑}{}_{\mathrm{n,m}}\text{→∑}{}_{\mathrm{n,m}}$

,

$\text{(H,(A,B))?(HAH}{}^{-1}\text{,HB)}$

Those canonical forms are obtained as an application of a more general method to select canonical elements M_c in the orbits $\text{O}{}_{\mathrm{M}}$ of a matrix group G acting on a set of matrices $\text{M}\text{?}\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$. We define a total order (?) on $\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$, different from the lexicographic order ^l? [0^l?x ? x <0, but $\text{0?x≠0 for x∈}\text{R}\text{]}$ and consider normalized $\text{O}{}_{\mathrm{M}}$-elements with a minimal number of parameters:

$\text{min{}\text{M}\text{?}\text{∈}\text{O}{}_{\mathrm{M}}\text{:}\text{M}\text{?}\text{normalized}}$

It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method.     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

More bounds for elgenvalues using traces

Let the n × n complex matrix A have complex eigenvalues λ₁,λ₂,…λ_n. Upper and lower bounds for Σ(Reλ_i)² are obtained, extending similar bounds for Σ|λ_i|² obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A¹A, B², C², and D², where $\text{B}\text{=}\text{1}\text{2}\text{(}\text{A}\text{+}\text{A}{}^1\text{)}$, $\text{C}\text{=}\text{1}\text{2}\text{(}\text{A}\text{?}\text{A}{}^1\text{) /i}$, and $\text{D}\text{=}\text{AA}{}^1\text{?}\text{A}{}^1\text{A}$, and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].     

Powers of matrices and idempotency

In this paper, we establish the following results: Let A be a square matrix of rank r. Then (a) $\text{(A+A}{}^1\text{)}\text{2}$ is idempotent of rank r, and tr_rA (defined as the sum of the principal minors of order r in A) is one iff A is Hermitian idempotent. (b) A^s=A^t for some positive integers s≠t, and trA=rankA iff A is idempotent. (c) $\text{A(A1A)}{}^{\mathrm{s}}\text{= A(AA}{}^1\text{)}{}^{\mathrm{t}}$ for some integers s≠t iff $\text{AA}{}^1\text{=A}{}^1\text{A}$ is idempotent, while $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{= A(AA}{}^1\text{)}{}^{\mathrm{s}}$ for some integers s≠0 iff $\text{AA}{}^1\text{=A}{}^1\text{A}$. (d) $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{=A}{}^1\text{(AA}{}^1\text{)}{}^{\mathrm{t}}$ for some integers s≠t and rankA=trA iff A is Hermitian idempotent, while $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{= A}{}^1\text{(AA}{}^1\text{)}{}^{\mathrm{s}}$ for some integer s iff A is Hermitian. Here $\text{A}{}^1$ indicates the conjugate transpose of A, and P^-α is defined iff (P⁺)^α=(P^α)⁺ for all positive integers α and P⁺ is the Moore-Penrose inverse of P.     

Derivations on a C1-algebra and its double dual

Let A be a C¹-algebra and X a Banach A-module. The module action of A on X gives rise to module actions of $\text{A}{}^7$ on $\text{X}{}^1$ and $\text{X}{}^7$, and derivations of A into X (resp. $\text{X}{}^1$) extend to derivations of $\text{A}{}^7$ into $\text{X}{}^7$ (resp. $\text{X}{}^1$). If A is nuclear, and X is a dual Banach A-module with $\text{X}{}^1$ weakly sequentially complete, then every derivation of A into X is inner. Under the same hypothesis on A, the extension to the finite part of $\text{A}{}^7$ of any derivation of A into any dual Banach A-module is inner, as are all derivations of A into $\text{A}{}^1$. Every derivation of a semifinite von Neumann algebra into its predual is inner.     

Brownian motion,Lp properties of Schrödinger operators and the localization of binding

We use Brownian motion ideas to study Schrödinger operators $\text{H =}\text{built?1}\text{2}\text{Δ + V}\text{on}\text{L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{v}}\text{)}$. In particular: (a) We prove that lim_t→∞t^?1In ∥ e^?tH ∥_p,p is p-independent for a very large class of V's where ∥ A ∥_p,p = norm of A as an operator from L^pto L^p. (b) For v ? 3 and $\text{V ? L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; ?\; ?</mtext>}}\text{∩ L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; +\; ?</mtext>}}$, we show that sup ∥ e^?tH ∥_∞,∞ < ∞ if and only if H has no negative eigenvalues or zero energy resonances. (c) We relate the “localization of binding” recently noted by Sigal to Brownian hitting probabilities.     

The KMS condition and spectral passivity in group duality

Let (A, G, α) be a C¹-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in $\text{G}\text{?}$ and a finite constant κ such that $\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}{}^1\text{x}{}_{\mathrm{i}}\text{) ? κ}\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}{}^1\text{)}$ whenever x_i lies in a spectral subspace $\text{R}{}^{\mathrm{\alpha }}\text{(Ω}{}_{\mathrm{i}}\text{)}$, where $\text{Ω}{}_1\text{+ … + Ω}{}_{\mathrm{n}}\text{? Ω}$. This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G.     

Continuity of positive functionals on topological 1-algebras

It is shown that the set $\text{C}$_{m × n} of complex m × n matrices forms a lower semilattice under the partial ordering A ? B defined by $\text{A}{}^1\text{A = A}{}^1\text{B,}{}^1\text{AA}{}^1\text{= BA}{}^1\text{,}\text{where}\text{A}{}^1$ denotes the conjugate transpose of A. As a special case of a result for division rings, it is further shown that, over any field F, form = n = 2 and any proper involution 1 of F_{2 × 2}, the corresponding intersections A ∩ B all exist.     

On the fair coin tossing process

Let Ω = {1, 0} and for each integer n ≥ 1 let $\text{Ω}{}_{\mathrm{n}}\text{= Ω × Ω × … × Ω}$ (n-tuple) and $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{= {(a}{}_1\text{, a}{}_2\text{, …, a}{}_{\mathrm{n}}\text{)|(a}{}_1\text{, a}{}_2\text{, … , a}{}_{\mathrm{n}}\text{) ? Ω}{}_{\mathrm{n}}\text{and}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{= k}}$ for all k = 0,1,…,n. Let {Y_m}_m≥1 be a sequence of i.i.d. random variables such that $\text{P(Y}{}_1\text{= 0) = P(Y}{}_1\text{= 1) =}\text{1}\text{2}$. For each A in $\text{Ω}{}_{\mathrm{n}}$, let T_A be the first occurrence time of A with respect to the stochastic process {Y_m}_m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in $\text{Ω}{}_{\mathrm{n}}$, there is an element B in $\text{Ω}{}_{\mathrm{n}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$. This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element B also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$; (II) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A = (a₁, a₂,…,a_n) in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element C also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_A is less than T_C is greater than $\text{1}\text{2}$ if n ≠ 2m or n = 2m but a_i = a_{i + 1} for some 1 ≤ i ≤ n?1. These new results provide us with a better and deeper understanding of the fair coin tossing process.     

Uniqueness of complex representing measures on the Choquet boundary

Let A be a linear subspace of complex C(X) which separates points and contains the constants. Hustad has shown that to each complex linear functional L in $\text{A}{}^1$ there corresponds a complex regular Borel measure μ “supported by” the Choquet boundary ?A of A which represents L and satisfies ∥ μ ∥ = ∥ L ∥. We give a number of conditions on the dual ball of A which are necessary and sufficient for each L in $\text{A}{}^1$ to be represented by a unique such measure μ.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

A fundamental theorem on lambda-matrices with applications —II. Difference equations with constant coefficients

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix $\text{L(λ) =}\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{λ}{}^{\mathrm{i}}$ where the A_i are n×n complex matrices and detA_l ≠ 0. In this paper general solutions are formulated for difference equations of the form $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{u}{}_{\mathrm{r\; +\; i}}\text{= ?}{}_{\mathrm{\gamma }}\text{, r = 1, 2,…}$. The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{S}{}^{\mathrm{i}}\text{= 0}$ for matrix S.     

Radial bounds for perturbations of elliptic operators

Elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$, α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}$ or a quotient space $\text{R}{}^{\mathrm{n}}\text{R}{}^{\mathrm{n}}\text{U}{}_{\mathrm{\alpha }}\text{, U}{}_{\mathrm{\alpha }}\text{? R}{}^{\mathrm{n}}$ are considered. It is shown that the L^p-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L¹ radial convolution kernel. Some consequences are: A can be closed in all L^p (1 ? p ? ∞), and is essentially self-adjoint in L² if it is symmetric; A generates an analytic semigroup e^?tA in the right half plane, strongly L^p and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability $\text{φ(εA)?→}{}_{\mathrm{\epsilon \; \to \; 0}}\text{φ(0) ?}$, with φ analytic, is proved for $\text{? ? L}{}^{\mathrm{p}}$, with convergence in L^p and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients.     

Zeta matrices of elliptic curves

Let $\text{O =}\text{lim}{}_{\mathrm{n}}\text{Z/p}{}^{\mathrm{n}}\text{Z}$, let $\text{A}\text{= O[g}{}_2\text{, g}{}_3\text{]}{}_{\mathrm{\Delta }}$, where g₂ and g₃ are coefficients of the elliptic curve: Y² = 4X³ ? g₂X ? g₃ over a finite field and Δ = g₂³ ? 27g₃² and let $\text{B}\text{=}\text{A}\text{[X, Y]}\text{(Y}{}^2\text{? 4X}{}^3\text{+ g}{}_2\text{X + g}{}_3\text{)}$. Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free $\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q}$-module $\text{H}{}^1\text{(X,}\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$. Main results are; Theorem 1.1: X²dY and YdX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{, Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$; Theorem 1.2: YdX, X²dY, Y^?1dX, Y^?2dX and XY^?2dX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{? (Y = 0), Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$, where $\text{X}$ is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.     

Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations

Let A(x,ε) be an n×n matrix function holomorphic for |x|?x₀, 0<ε?ε₀, and possessing, uniformly in x, an asymptotic expansion $\text{A(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{A}{}_{\mathrm{r}}\text{(x) ε}{}^{\mathrm{r}}$, as ε→0⁺. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion $\text{P(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{P}{}_{\mathrm{r}}\text{(x)ε}{}^{\mathrm{r}}$, as ε→0⁺, is constructed, such that the transformation y = P(x,ε)z takes the differential equation $\text{ε}{}^{\mathrm{h}}\text{dy}\text{dx}\text{= A(x,ε)y,h}$ a positive integer, into $\text{ε}{}^{\mathrm{h}}\text{dz}\text{dx}\text{= B(x,ε)z}$, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.