Scalar Equations and First-Order Systems

A scalar second-order equation and a rank-two first-order system carry the same local solution count, but they retain different auxiliary data. The scalar equation singles out a cyclic component; the system makes gauges, parallel transport, and determinant lines transparent. Moving safely between the two requires keeping that extra structure visible.

Throughout this page, $U$ is a connected domain on which the displayed coefficients are holomorphic. Singular points are removed from $U$ until their local analysis is explicitly restored.

From a scalar equation to a companion system

Start with

$$y''+p(z)y'+q(z)y=0.$$

For the state vector

$$Y= \begin{pmatrix} y\\ y' \end{pmatrix},$$

the equation is equivalent to

$$Y'=A_{\mathrm{comp}}Y, \qquad A_{\mathrm{comp}} = \begin{pmatrix} 0 & 1\\ -q & -p \end{pmatrix}.$$

If $y_1$ and $y_2$ are independent scalar solutions, the corresponding fundamental matrix is

$$\Phi= \begin{pmatrix} y_1 & y_2\\ y_1' & y_2' \end{pmatrix}.$$

Its determinant is the scalar Wronskian,

$$\det\Phi =\Wr[y_1,y_2] =y_1y_2'-y_1'y_2.$$

The companion construction is canonical only after the scalar dependent variable has been chosen. A scalar gauge, a different component of a system, or a different cyclic vector produces a different companion matrix.

Existence and normalized fundamental matrices

For a holomorphic matrix $A:U\to M_r(\mathbb C)$, the initial-value problem

$$\Phi'(z)=A(z)\Phi(z), \qquad \Phi(z_0)=\Phi_0$$

has a unique local holomorphic solution. If $\Phi_0$ is invertible, then $\Phi$ remains invertible wherever it is continued inside $U$.

On a simply connected $U$, the normalization

$$\Phi(z_0)=I$$

defines a single-valued fundamental matrix throughout $U$. On a non-simply-connected domain, continuation around a loop can return $\Phi M_\gamma$ instead; this is monodromy, not a failure of local existence.

Any two fundamental matrices for the same system on a connected overlap differ by a constant matrix on the right. Indeed, if

$$\widetilde\Phi=\Phi H(z),$$

then

$$\widetilde\Phi' =A\widetilde\Phi+\Phi H'.$$

Thus $\widetilde\Phi'=A\widetilde\Phi$ exactly when $H'=0$.

Frame versus horizontal transport

The normalization $\Phi(z_0)=I$ chooses a frame in one fiber. The differential equation transports that frame. Changing the initial frame acts on the right; changing the dependent-variable frame by a gauge acts on the left.

Abel–Liouville identities

Jacobi’s determinant formula gives

$$\frac{\dd}{\dd z}\det\Phi =\operatorname{tr} \left( \operatorname{adj}(\Phi)\Phi' \right).$$

Using $\Phi'=A\Phi$ and $\operatorname{adj}(\Phi)=\det(\Phi)\Phi^{-1}$ yields

$$(\det\Phi)' =\operatorname{tr}(A)\det\Phi.$$

Consequently,

$$\det\Phi(z) =\det\Phi(z_0) \exp\left( \int_{z_0}^{z}\operatorname{tr}A(\zeta)\,\dd\zeta \right)$$

on a chosen continuation path.

For the companion matrix, $\operatorname{tr}A_{\mathrm{comp}}=-p$, so

$$\Wr[y_1,y_2](z) =\Wr[y_1,y_2](z_0) \exp\left( -\int_{z_0}^{z}p(\zeta)\,\dd\zeta \right).$$

This is Abel’s identity. It has three immediate consequences:

1. a Wronskian that is nonzero at one ordinary point is nonzero throughout the connected regular domain;
2. a scalar second-order solution pair cannot become dependent at an isolated ordinary point;
3. the determinant holonomy is controlled by the trace connection.

For a closed loop $\gamma$ based at $z_0$,

$$\det M_\gamma =\exp\left( \oint_\gamma\operatorname{tr}A(z)\,\dd z \right).$$

In the scalar companion form this becomes

$$\det M_\gamma =\exp\left( -\oint_\gamma p(z)\,\dd z \right).$$

The period depends only on $[\gamma]\in H_1(U,\mathbb Z)$. Conversely, recovering the period from $\det M_\gamma$ determines it only modulo $2\pi\ii\mathbb Z$.

Gauge transformations and basis changes

Let

$$\widetilde Y=G(z)Y, \qquad G(z)\in GL(r,\mathbb C).$$

Then

$$\widetilde Y' =\widetilde A\,\widetilde Y, \qquad \widetilde A =GAG^{-1}+G'G^{-1}.$$

This is a gauge transformation. It changes the matrix representing the connection, and it may change the visible singularities if $G$ or $G^{-1}$ is meromorphic rather than holomorphic.

By contrast, a constant change of solution basis is

$$\widetilde\Phi=\Phi H, \qquad H\in GL(r,\mathbb C).$$

It leaves $A$ unchanged. If continuation is represented by $\Phi^\gamma=\Phi M_\gamma$, then

$$\widetilde M_\gamma =H^{-1}M_\gamma H.$$

For two canonical bases with $\Phi_\beta=\Phi_\alpha C_{\alpha\beta}$, independent changes $H_\alpha$ and $H_\beta$ give

$$\widetilde C_{\alpha\beta} =H_\alpha^{-1}C_{\alpha\beta}H_\beta.$$

The distinction is visible in the side on which the matrix acts:

Transformation	Action on $\Phi$	Effect on $A$
Gauge	$G(z)\Phi$	$A\mapsto GAG^{-1}+G'G^{-1}$
Solution basis	$\Phi H$	None
Analytic continuation	$\Phi\mapsto\Phi M_\gamma$	None

Coordinate changes

Under a locally biholomorphic change $z=z(w)$, the system becomes

$$\frac{\dd Y}{\dd w} =\frac{\dd z}{\dd w}A(z(w))Y.$$

Equivalently, the matrix-valued connection one-form

$$A(z)\,\dd z$$

is pulled back to the $w$-coordinate. A biholomorphic coordinate change preserves the local analytic type. A ramified map or a singular parameter-dependent scaling can change formal slopes and must be analyzed as part of the problem, not treated as an innocuous relabeling.

Recovering a scalar equation from a system

Consider

$$\begin{pmatrix} u\\ v \end{pmatrix}' = \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} u\\ v \end{pmatrix}.$$

Where $b\neq0$, the first row gives

$$v=\frac{u'-au}{b}.$$

Substitution into the second row produces

$$u''+p_u u'+q_u u=0,$$

where

$$\begin{aligned} p_u &=-a-d-\frac{b'}b,\\ q_u &=-a' +a\frac{b'}b -bc+ad. \end{aligned}$$

The formula is easily checked by rebuilding the companion system. Its important feature is the logarithmic derivative $b'/b$. Suppose the system is holomorphic and $b$ has an isolated zero of order $m$ at $z_*$. Then $p_u$ has residue $-m$, while $q_u$ has at most a simple pole. The induced scalar equation has indicial polynomial

$$\rho(\rho-m-1),$$

so its exponents are $0$ and $m+1$. Moreover, for any system fundamental matrix $\Phi$,

$$\Wr[u_1,u_2]=b\,\det\Phi.$$

Both component solutions extend holomorphically across $z_*$, and their scalar monodromy is trivial. Every such isolated zero of $b$ is therefore an apparent regular singularity of this scalar presentation, even though it is an ordinary point of the system.

The system and scalar equation are equivalent only on the region where the chosen cyclic vector is valid, together with the extension data across its zeros.

If $b$ vanishes identically, the component $u$ is not a cyclic vector for this presentation. One must choose another component or a more general linear functional on the rank-two bundle.

The determinant line and traceless reduction

For a rank-two flat bundle $E$, its determinant line is

$$\det E=\bigwedge^2E.$$

In a local frame its induced scalar equation is

$$w'=\operatorname{tr}(A)w,$$

and its holonomy around $\gamma$ is $\det M_\gamma$.

Now decompose

$$A=A_0+\frac12\operatorname{tr}(A)I, \qquad \operatorname{tr}(A_0)=0.$$

Locally choose

$$g(z) =\exp\left( -\frac12\int^z\operatorname{tr}A(\zeta)\,\dd\zeta \right).$$

The scalar gauge $\widetilde Y=gY$ gives

$$\widetilde A =A+\frac{g'}g I =A_0.$$

This proves a local traceless reduction on a simply connected regular neighborhood. Globally, $g$ can be multivalued. Two related global questions must be distinguished:

• A single-valued global $GL(2)$ gauge with traceless transformed connection requires a horizontal trivialization of the determinant connection. In the present trivial-bundle setting, this is equivalent to trivial determinant holonomy.
• A scalar gauge $gI$ additionally chooses a square root of that horizontal determinant gauge, and the square root must itself be single-valued. This is analogous to, but distinct from, the theta characteristic $K^{1/2}$ used for coordinate-global scalar oper form.

Thus local trace-free matrices need not glue as a gauge of the original $GL(2)$ system. An $SL(2)$ lift of the projectivized local system is a different construction: one twists by a rank-one character $\chi$ satisfying $\chi^2=(\det\rho)^{-1}$. That twist is not a gauge transformation of the original global system.

A punctured-plane example

On $\mathbb C^\times$, take

$$A(z)=\frac{a}{z}I.$$

A local scalar trace-removing gauge is

$$g(z)=z^{-a}.$$

It is single-valued on $\mathbb C^\times$ only when $a\in\mathbb Z$. The determinant holonomy is

$$\det M_0=\ee^{4\pi\ii a}.$$

When $2a\in\mathbb Z$, the determinant local system is trivial even if the scalar square root is not. For example, at $a=1/2$ the single-valued gauge

$$G(z)= \begin{pmatrix} z^{-1} & 0\\ 0 & 1 \end{pmatrix}$$

produces

$$\widetilde A = \frac1{2z} \begin{pmatrix} -1 & 0\\ 0 & 1 \end{pmatrix}.$$

The example separates determinant trivialization from the stronger demand that the trace be removed by a scalar square root.

Its projective monodromy is trivial for every $a$. The condition $2a\in\mathbb Z$ characterizes when the original $GL(2)$ system is gauge-equivalent through a single-valued gauge to a trace-free one; it is not an obstruction to lifting the trivial projective monodromy.

Holomorphic dependence on parameters

Let $\lambda$ range in a parameter domain $\Lambda$, and assume $A(z,\lambda)$ is holomorphic on $U\times\Lambda$. Fix $z_0\in U$ and normalize

$$\Phi(z_0,\lambda)=I.$$

On a simply connected $U$, the resulting fundamental matrix is jointly holomorphic in $(z,\lambda)$. Differentiating with respect to $\lambda$ gives

$$(\partial_\lambda\Phi)' =(\partial_\lambda A)\Phi +A(\partial_\lambda\Phi).$$

Set

$$B_\lambda =\Phi^{-1}\partial_\lambda\Phi.$$

Then

$$B_\lambda' =\Phi^{-1}(\partial_\lambda A)\Phi, \qquad B_\lambda(z_0)=0,$$

and hence

$$\partial_\lambda\Phi(z,\lambda) =\Phi(z,\lambda) \int_{z_0}^{z} \Phi(\zeta,\lambda)^{-1} (\partial_\lambda A)(\zeta,\lambda) \Phi(\zeta,\lambda)\,\dd\zeta.$$

This variation formula is the linear starting point for deformation theory. Its hypotheses matter:

• the integration path must remain in a common regular domain;
• moving singularities require a domain or coordinate trivialization;
• a canonical Frobenius or sectorial basis can be meromorphic or branched in $\lambda$ even when the normalized initial-value matrix is holomorphic;
• differentiating monodromy also differentiates the continuation problem and any moving generators.

A normalization workflow

Before comparing two scalar or system presentations, record:

1. the domain, base point, and excluded parameter values;
2. the scalar cyclic vector or system frame;
3. the gauge and coordinate transformations, including their branches;
4. the fundamental-matrix normalization;
5. the Wronskian or determinant convention;
6. the continuation path and loop composition;
7. any zeros or poles of the cyclic-vector coefficient;
8. whether the claimed equivalence is local, meromorphic, or global.

This ledger turns a formal conversion into a reproducible equivalence statement.

Common pitfalls

Multiplying on the wrong side. A gauge acts on a column solution from the left; a change among solution columns acts on a fundamental matrix from the right. Confusing them produces the wrong connection-matrix covariance.

Assuming the Wronskian is always constant. It is constant in scalar normal form, but in a general scalar equation it evolves by Abel’s identity.

Eliminating a component through a zero. The formula $v=(u'-au)/b$ is valid only where $b\neq0$. Zeros of $b$ can create apparent singularities and must be patched with another cyclic vector.

Globalizing the trace-removing exponential. A locally defined primitive of $\operatorname{tr}A$ need not be single-valued. Determinant holonomy and its square root are genuine global data.

Exercises

1. Abel from Jacobi. Derive $(\det\Phi)'=\operatorname{tr}(A)\det\Phi$ without assuming that $A$ is diagonalizable. Specialize to the scalar companion system.

Solution

Where $\Phi$ is invertible, Jacobi’s formula gives

$$(\det\Phi)' =\det\Phi\, \operatorname{tr}(\Phi^{-1}\Phi').$$

Since $\Phi'=A\Phi$,

$$\operatorname{tr}(\Phi^{-1}A\Phi) =\operatorname{tr}A$$

by cyclicity of trace. No spectral decomposition of $A$ is used. For the companion matrix, $\operatorname{tr}A=-p$, so $\Wr'=-p\,\Wr$.

2. Cyclic-vector reduction. Starting from the rank-two system with $b\neq0$, derive the displayed $p_u$ and $q_u$. Check the result on the companion matrix.

Solution

Differentiate $u'=au+bv$:

$$u'' =a'u+au' +b'v+b(cu+dv).$$

Substitute $v=(u'-au)/b$ and collect terms:

$$\begin{aligned} u'' &=\left( a+d+\frac{b'}b \right)u'\\ &\quad+ \left( a' -a\frac{b'}b +bc-ad \right)u. \end{aligned}$$

Moving both terms to the left yields the stated coefficients. For $a=d=0$, $b=1$, and $c=-q$, one obtains $p_u=0$ and $q_u=q$. For the full companion matrix $d=-p$, the result is $p_u=p$ and $q_u=q$.

3. Scalar versus general trace removal. For $A=(a/z)I$ on $\mathbb C^\times$, determine when the scalar gauge $z^{-a}I$ is single-valued and when some single-valued holomorphic gauge $G:\mathbb C^\times\to GL(2,\mathbb C)$ can make the system traceless.

Solution

The scalar gauge is single-valued exactly when $a\in\mathbb Z$. A general gauge satisfies

$$\operatorname{tr}\widetilde A =\frac{2a}{z} +\frac{\dd}{\dd z}\log\det G.$$

Thus one needs $\det G$ proportional to $z^{-2a}$. A single-valued holomorphic and invertible choice on $\mathbb C^\times$ exists exactly when $2a\in\mathbb Z$. At half-integral $a$, a diagonal gauge can remove the trace although no single-valued scalar square-root gauge exists.

4. Parameter variation. Verify the formula for $\partial_\lambda\Phi$ and use it for a constant family $A(z,\lambda)=\lambda A_0$.

Solution

For the constant family normalized at $z_0$,

$$\Phi(z,\lambda) =\exp\bigl(\lambda(z-z_0)A_0\bigr).$$

Because $A_0$ commutes with $\Phi$,

$$\begin{aligned} \partial_\lambda\Phi &=\Phi \int_{z_0}^{z}A_0\,\dd\zeta\\ &=(z-z_0)\Phi A_0, \end{aligned}$$

which agrees with direct differentiation of the matrix exponential.

References

• G. Teschl, Ordinary Differential Equations and Dynamical Systems, for fundamental matrices, determinant identities, and analytic dependence.
• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw–Hill, 1955, for the classical theory of linear systems and parameter dependence.
• K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé, for Fuchsian systems, cyclic vectors, monodromy, and deformation theory.
• P. Deligne, Équations différentielles à points singuliers réguliers, for flat bundles, determinant connections, and regular singularities.