Analytic function of a matrix

1 Definition and Construction

1.1 Power series definition \(f(A)=\sum_{k\ge 0} a_k A^k\)

For a square complex matrix \(A\in\mathbb{C}^{n\times n}\), an analytic scalar function \(f\) can be expressed near a point as a power series \[ f(z)=\sum_{k\ge 0} a_k (z-z_0)^k, \] or, after shifting, simply \(f(z)=\sum_{k\ge 0} a_k z^k\) in the origin-centered case. The matrix function is then defined by substituting the matrix for the variable: \[ f(A)=\sum_{k\ge 0} a_k A^k. \] This definition is natural because it preserves the polynomial/algebraic operations used to build operator functions and matches the scalar rule when \(A\) is replaced by a scalar multiple of the identity.

If the series is centered at \(z_0\), the same idea gives \(f(A)=\sum_{k\ge 0} a_k (A-z_0 I)^k\).

1.2 Convergence and radius of validity

Power series for matrices converge according to the same coefficients but with the matrix substituted for the scalar. In practice, convergence is governed by the operator norm: if \(\| \cdot \|\) is any submultiplicative matrix norm, then

\[

\sum_{k\ge 0} \|a_k A^k\|

\] converges whenever the scalar series converges after replacing \(z\) by a quantity that reflects the size of \(A\). A standard criterion involves the spectral radius \(\rho(A)\): if the scalar power series for \(f\) has radius of convergence \(R\), then the matrix series converges at least whenever \(\rho(A)<R\). The precise boundary depends on the function’s analytic continuation and the chosen representation, but the analytic functional calculus ensures a consistent definition on the domain where \(f\) is holomorphic.

1.3 Polynomial approximation viewpoint

Analytic functions admit approximation on compact sets by polynomials (for example, via Runge’s theorem or other complex-analytic approximation results). This yields an alternative construction: one approximates \(f\) uniformly by polynomials \(p_m\) on a region containing the spectrum \(\sigma(A)\), and defines \[ f(A)=\lim_{m\to\infty} p_m(A). \] This viewpoint clarifies why matrix functions often satisfy identities mirroring scalar function properties: many scalar facts are first proved for polynomials and then passed to limits.

1.4 Holomorphic functional calculus (contour integral definition)

A central, representation-independent definition uses contour integration. If \(f\) is holomorphic on an open set containing \(\sigma(A)\), then \[ f(A)=\frac{1}{2\pi i}\int_\Gamma f(z)\,(zI-A)^{-1}\,dz, \] where \(\Gamma\) is a positively oriented simple closed contour lying in the holomorphic domain of \(f\) and enclosing \(\sigma(A)\). The matrix resolvent \((zI-A)^{-1}\) is analytic in \(z\) away from the spectrum, and the contour integral produces a well-defined matrix that depends only on the values of \(f\) on the enclosed region.

This approach also connects directly to Cauchy-type formulas and makes differentiation and other structural results more transparent.

1.5 Independence of the chosen representation

The defining formulas agree when they are simultaneously valid. If \(f\) has a convergent power-series expansion around a point in the holomorphic region, then the power-series substitution \(f(A)=\sum a_k A^k\) matches the contour integral definition. Similarly, polynomial approximation limits agree with the integral formula. This independence is a key feature: \(f(A)\) is determined by the holomorphic function \(f\) rather than by the particular algebraic expression used to describe it.

2 Spectral and Jordan Structure

2.1 Eigenvalues and the spectrum of \(f(A)\)

The spectrum of \(f(A)\) is constrained by the spectrum of \(A\). In general, if \(f\) is holomorphic on a neighborhood of \(\sigma(A)\), then every eigenvalue \(\mu\) of \(f(A)\) must be of the form \(\mu=f(\lambda)\) for some \(\lambda\in\sigma(A)\). More precisely, the spectral mapping theorem states this relationship exactly in terms of spectra.

Eigenvectors need not be preserved when \(A\) is nonnormal or nondiagonalizable, but the locations of eigenvalues of \(f(A)\) follow the scalar mapping through \(f\).

2.2 Jordan canonical form approach

When \(A\) is expressed in Jordan canonical form, the action of \(f\) becomes explicit and reveals how nontrivial nilpotent parts affect the result.

If \(A=SJS^{-1}\) with \(J\) a direct sum of Jordan blocks, then similarity invariance gives \[ f(A)=S\,f(J)\,S^{-1}, \] so it suffices to understand \(f\) applied to a Jordan block.

2.2.1 Evaluation on Jordan blocks

Consider a Jordan block \(J=\lambda I + N\), where \(N\) is nilpotent (a shift matrix with \(N^m=0\) for some \(m\)). For holomorphic \(f\), one obtains \[ f(J)=\sum_{r=0}^{m-1} \frac{f^{(r)}(\lambda)}{r!}\,N^r. \] Thus, applying \(f\) to a Jordan block is equivalent to a truncated Taylor expansion in the nilpotent direction.

2.2.2 Role of derivatives \(f^{(k)}(\lambda)\) for nilpotent parts

If \(A\) has a Jordan block of size exceeding one, the nilpotent part contributes higher-order terms involving derivatives of \(f\) at the corresponding eigenvalue. This explains why \(f(A)\) can carry information beyond merely substituting eigenvalues: the size and structure of Jordan blocks determine which derivatives appear and how they populate superdiagonals within \(f(J)\).

In the diagonalizable case (all Jordan blocks are size one), only zeroth derivatives are needed, and \(f(A)\) reduces to \(f\) applied to eigenvalues in the usual spectral decomposition.

2.3 Spectral mapping theorem

For holomorphic \(f\) defined on a neighborhood of \(\sigma(A)\), \[ \sigma(f(A))=f(\sigma(A)). \] This equality captures both inclusion and exactness: the spectrum of the matrix function is precisely the image of the original spectrum under \(f\). The statement extends to more refined notions involving approximate point spectra and other spectral sets, but the core theorem is the direct image rule.

3 Basic Properties and Algebraic Rules

3.1 Commutativity: \(f(A)A=Af(A)\)

Matrix functions defined via polynomials or the functional calculus commute with the underlying matrix. If \(f\) is analytic on a domain containing \(\sigma(A)\), then \(f(A)\) commutes with \(A\): \[ f(A)A = A f(A). \] For polynomial approximations, this is immediate because polynomials in \(A\) commute with \(A\); passing to the limit preserves commutativity.

3.2 Linearity and product rules

The mapping \(f\mapsto f(A)\) is linear when functions are added or scaled: \[ (\alpha f+\beta g)(A)=\alpha f(A)+\beta g(A). \] Product behavior is similarly compatible with analytic multiplication: \[ (fg)(A)=f(A)\,g(A), \] as long as the holomorphic domains permit evaluation. This reflects that the functional calculus respects algebraic operations in the Banach-algebra sense for holomorphic functions.

3.3 Compatibility with polynomials

If \(p\) is a polynomial, then the matrix function agrees with the usual evaluation: \[ p(A)=\sum_k c_k A^k. \] Moreover, if \(f\) is holomorphic and \(p_m\) approximates \(f\) on a set enclosing \(\sigma(A)\), then \(p_m(A)\) converges to \(f(A)\). This compatibility is often used as the starting point for proofs: establish identities for polynomials and extend them to holomorphic functions by approximation.

3.4 Similarity invariance: \(f(SAS^{-1})=Sf(A)S^{-1}\)

Functional calculus is consistent under change of basis. For invertible \(S\), \[ f(SAS^{-1})=S f(A) S^{-1}. \] This property follows directly from the contour integral definition, since \((zI-SAS^{-1})^{-1}=S(zI-A)^{-1}S^{-1}\) and \(S\) and \(S^{-1}\) can be pulled through the integral. As a result, \(f(A)\) depends on \(A\) up to similarity, aligning with the role of Jordan and spectral structure.

3.5 Functional identities and resolvent-based formulas

Holomorphic functional calculus yields identities resembling scalar ones. A key example uses the resolvent: \[ (zI-A)^{-1} \] is itself an analytic matrix-valued function of \(z\). Many functional identities can be expressed through partial fraction decompositions or resolvent relations, and then integrated against \(f(z)\) using the contour integral. This method is especially effective for rational functions \(f\) that can be written in terms of resolvents and matrix inverses.

4 Analytic Dependence on Parameters

4.1 Matrix families \(A(t)\) and analytic \(f\)

Suppose \(A(t)\) is a matrix-valued map depending on a parameter \(t\) (for instance, real or complex near a point), and \(f\) is holomorphic on a domain containing \(\sigma(A(t))\) for \(t\) in a neighborhood. Under such assumptions, the composed map \[ t\mapsto f(A(t)) \] is typically continuous, and often differentiable, with regularity linked to both \(A(t)\) and the analytic structure of the functional calculus.

4.2 Differentiation of \(f(A(t))\) (Fréchet derivatives)

The derivative of \(f(A(t))\) with respect to a perturbation \(E\) of the matrix is captured by a Fréchet derivative \(DfA\). For analytic \(f\), this derivative is linear in \(E\) and describes the first-order response: \[

f(A+E)=f(A)+DfA+o(\|E\|).

\] When \(f\) is holomorphic and defined on a neighborhood of \(\sigma(A)\), the Fréchet derivative exists under broad conditions and can be expressed in terms of resolvents and contour integrals.

4.3 Derivative formulas via contour integrals

Using the contour integral representation, one may differentiate under the integral sign (subject to standard justification) to obtain formulas such as \ Df[A=\frac{1}{2\pi i}\int_\Gamma f(z)\,(zI-A)^{-1} E (zI-A)^{-1}\,dz. \] This expresses how perturbations propagate through the resolvent twice, reflecting the noncommutative nature of matrix multiplication. Higher derivatives can also be represented by repeated insertions of \(E\) in the resolvent factors.

4.4 Perturbation behavior and stability considerations

Because the functional calculus is nonlinear in general, small changes in \(A\) can induce changes in \(f(A)\) whose magnitude depends on both the function and the spectral configuration. Stability is influenced by factors such as:

• proximity of the spectrum to the singularities or branch points of \(f\),
• nonnormality of \(A\), which can amplify perturbations,
• conditioning of eigenvalues/Jordan blocks when diagonalization is ill-behaved.

These considerations motivate error bounds in numerical algorithms and help interpret sensitivity in applications.

5 Special Analytic Matrix Functions

5.1 Matrix exponential \(\exp(A)\)

The matrix exponential is defined by the power series \[ \exp(A)=\sum_{k=0}^\infty \frac{1}{k!}A^k, \] which converges for all matrices. The exponential is the central example of an analytic matrix function, and it satisfies familiar differential identities analogous to the scalar case, such as:

• \(\exp(A+B)=\exp(A)\exp(B)\) holds when \(A\) and \(B\) commute,
• \(\frac{d}{dt}\exp(tA)=A\exp(tA)=\exp(tA)A\).

Through Jordan form, \(\exp(J)\) can be computed explicitly via truncated Taylor polynomials in the nilpotent part.

5.2 Matrix logarithm \(\log(A)\) and branch choices

A matrix logarithm exists only under conditions that prevent crossing the branch cut. For \(A\) with no eigenvalues on a chosen branch cut, one defines \(\log(A)\) via the holomorphic functional calculus: \[ \log(A)=\frac{1}{2\pi i}\int_\Gamma \log(z)\,(zI-A)^{-1}\,dz, \] where \(\log(z)\) is a chosen branch of the scalar logarithm. Different branches produce different values of \(\log(A)\) by integer multiples of \(2\pi i\) on suitable invariant subspaces.

When \(A\) is invertible and has a spectrum contained in a simply connected domain avoiding the nonpositive real axis (for the principal branch), the principal logarithm is well-defined and analytic in that region.

5.3 Matrix resolvent \((zI-A)^{-1}\) as an analytic function

The resolvent is a fundamental example of an analytic matrix-valued function of the complex parameter \(z\). For \(z\notin\sigma(A)\), the inverse \((zI-A)^{-1}\) exists and depends analytically on \(z\). It underlies the contour integral definition and provides a direct route to compute or bound \(f(A)\) for many choices of \(f\).

The resolvent identity also offers algebraic relations helpful for both theory and computation.

5.4 Trigonometric and hyperbolic matrix functions

Trigonometric and hyperbolic functions of a matrix are defined by their scalar power series, such as \[ \sin(A)=\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}A^{2k+1},\quad \cos(A)=\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}A^{2k}, \] and similarly for \(\sinh\) and \(\cosh\). These functions satisfy matrix analogues of classical identities, with caveats about commutativity when combining functions of different matrices. Many identities can be derived using the exponential representation (e.g., \(\sin(A)\) and \(\cos(A)\) expressed via \(\exp(\pm iA)\)).

5.5 Fractional powers \(A^\alpha\)

Fractional powers extend the idea of \(z^\alpha\) to matrices and require careful branch selection. For a holomorphic choice of \(z^\alpha\) on a domain containing \(\sigma(A)\), one defines \[ A^\alpha = f(A) \] with \(f(z)=z^\alpha\) taken on that branch. Existence and analytic dependence require that the spectrum avoids the branch cut and remains in the holomorphic region for the selected branch. The resulting \(A^\alpha\) generally depends on the branch, and it satisfies \(A^\alpha A^\beta = A^{\alpha+\beta}\) when the functional calculus is applied consistently on a compatible domain.

6 Computation and Practical Evaluation

6.1 Using polynomial/rational approximations (e.g., Padé)

In numerical practice, computing \(f(A)\) often relies on approximating \(f\) by functions that are easy to evaluate on matrices. Polynomial approximations approximate \(f\) on a region containing the spectrum. Rational approximations are especially useful because they can be evaluated using solutions to linear systems involving \(zI-A\), similar to resolvent evaluation.

Padé approximants are common for the exponential and related functions: \(\exp(A)\) can be approximated by a rational function whose coefficients are chosen to match the Taylor series to high order.

6.2 Diagonalizable matrices and eigen-decomposition

If \(A\) is diagonalizable, then \(A=S\Lambda S^{-1}\) with \(\Lambda\) diagonal. The functional calculus becomes straightforward: \[ f(A)=S\,f(\Lambda)\,S^{-1}, \] where \(f(\Lambda)\) is the diagonal matrix formed by applying \(f\) to each eigenvalue. This reduces the problem to scalar evaluation plus similarity transformation.

However, numerical stability may suffer if \(S\) is ill-conditioned, even when diagonalization is exact in theory.

6.3 Jordan-form computation (conceptual and limitations)

Jordan form provides explicit formulas but is rarely used directly in computation because Jordan decomposition is numerically unstable: small perturbations of \(A\) can change Jordan blocks drastically. Still, the conceptual value remains high. It explains the derivative structure in \(f(J)\) and clarifies why functions of nondiagonalizable matrices involve higher-order terms tied to Jordan block sizes.

6.4 Numerical methods: scaling-and-squaring for exponentials

For \(\exp(A)\), a widely used approach is scaling-and-squaring. One computes \(\exp(A/2^m)\) accurately (using a rational approximation), then repeatedly squares the result: \[ \exp(A)=\left(\exp(A/2^m)\right)^{2^m}. \] Scaling reduces the effective norm of the matrix argument and improves approximation quality while controlling rounding errors.

6.5 Conditioning and numerical sensitivity

The numerical stability of computing \(f(A)\) depends on how sensitive \(f(A)\) is to perturbations in \(A\). Conditioning is influenced by:

• distance from the spectrum to singularities of \(f\) (e.g., branch points for logarithms),
• nonnormality, which can cause large transient amplification even if eigenvalues are stable,
• the growth of derivatives of \(f\) that appear in Jordan-block contributions.

Good algorithms incorporate these factors through scaling, adaptive approximations, and careful selection of branches for multivalued functions.

7 Theoretical Frameworks and Extensions

7.1 Cauchy integral formula for matrices

The contour integral definition of \(f(A)\) can be viewed as a matrix-valued analogue of the Cauchy integral formula. The resolvent \((zI-A)^{-1}\) plays the role of \(1/(z-\lambda)\) in scalar complex analysis. As a result, many complex-analytic properties translate into operator identities, enabling a systematic treatment of holomorphic functions of matrices.

7.2 Banach algebra perspective of functional calculus

Matrix functions fit naturally into the setting of Banach algebras. In this framework, \(f(A)\) is defined for elements \(A\) of a unital Banach algebra using a holomorphic functional calculus consistent with the spectrum defined in the algebraic sense. For finite-dimensional matrices, the Banach-algebra viewpoint streamlines proofs of algebraic properties like homomorphism behavior and similarity invariance.

7.3 Extension to non-analytic functional calculus (overview boundary)

Beyond the analytic category, one may define functions of matrices using weaker notions, such as:

• continuous functional calculus for normal matrices,
• spectral measure approaches (primarily in operator theory),
• calculus for functions that are not holomorphic, often requiring additional structure.

These extensions generally lose some properties enjoyed by analytic functional calculus, and they require different definitions and assumptions. The analytic theory remains the most systematic and broadly applicable within the matrix setting.

7.4 Connections to operator theory and semigroups

The exponential \(e^{tA}\) is central to operator theory and the study of dynamical systems. It links analytic matrix functions to semigroup behavior: for suitable matrices, the family \(\{e^{tA}\}_{t\ge 0}\) forms a semigroup under multiplication and solves linear differential equations with constant coefficients: \[ \frac{d}{dt}X(t)=AX(t). \] More generally, resolvents and functional calculus methods inform the construction and analysis of semigroups generated by operators, a theme that extends well beyond finite dimensions.

7.5 Relation to differential equations with matrix coefficients

Matrix functions arise naturally when solving differential equations where coefficients and initial conditions lead to expressions involving \(\exp(At)\). Additionally, logarithms and fractional powers appear in formulations for systems with delays, stability transformations, or fractional dynamics. In these contexts, ensuring that \(f(A)\) is well-defined and behaves continuously with respect to parameters is essential for qualitative analysis.

8 Examples and Worked Exercises

8.1 Computing \(f(A)\) for small matrices via power series

Take a small matrix \(A\) and an analytic function \(f\) with known Taylor coefficients. One may compute \(f(A)\) directly from \[ f(A)=\sum_{k\ge 0} a_k A^k \] until the remainder is below a desired tolerance. For \(2\times 2\) matrices, powers often stabilize quickly due to low-dimensional constraints, and the series may be accelerated using polynomial identities derived from the characteristic polynomial.

A typical exercise is to compute \(\sin(A)\) or \(\exp(A)\) for a matrix with small norm using a truncated series, then verify results by checking whether the defining differential identity holds numerically.

8.2 Example using contour integral for a rational function

Let \(f(z)=\frac{1}{z-\zeta}\) for some \(\zeta\) outside \(\sigma(A)\). The associated contour integral recovers the resolvent: \[ f(A)=\frac{1}{2\pi i}\int_\Gamma \frac{1}{z-\zeta}(zI-A)^{-1}\,dz. \] By residue calculus, the integral evaluates to \((\zeta I-A)^{-1}\). This illustrates how the general holomorphic functional calculus reduces to familiar matrix inverses for rational functions.

8.3 Example with a single Jordan block

Let \(J=\lambda I+N\) be a Jordan block of size \(m\) with nilpotent \(N\) satisfying \(N^m=0\). For an analytic function \(f\), the evaluation is \[ f(J)=\sum_{r=0}^{m-1}\frac{f^{(r)}(\lambda)}{r!}N^r. \] A worked exercise might specify \(m=3\) and choose a function like \(\exp\) or \(\log\) on a suitable domain, then compute \(f(J)\) explicitly by writing \(N^2\) and placing coefficients in the correct superdiagonal positions.

8.4 Example: resolvent expansion and geometric series

If \(A\) is such that \(\|A\|<	z	\), then the resolvent can be expanded as a geometric series:

\[ (zI-A)^{-1}=\frac{1}{z}\left(I-\frac{A}{z}\right)^{-1} =\frac{1}{z}\sum_{k=0}^\infty \left(\frac{A}{z}\right)^k. \] Using the contour integral definition with an integrand involving the resolvent, one can derive series representations for functions defined through the resolvent, and connect them to power-series definitions when both are valid.

8.5 Example: parameter-dependent matrix and differentiation

Consider a differentiable family \(A(t)\) and an analytic function \(f\). The Fréchet derivative formula gives a practical expression for the derivative at a point: \[

\frac{d}{dt}f(A(t))\Big	_{t=t_0}=Df[A(t_0)]\left(A'(t_0)\right),

\] with \ Df[A=\frac{1}{2\pi i}\int_\Gamma f(z)\,(zI-A)^{-1} E (zI-A)^{-1}\,dz. \]

An exercise may specify \(A(t)=A_0+tE\) and use the integral representation numerically (choosing a contour and discretizing the integral) to estimate \(\frac{d}{dt}f(A(t))\big	_{t=0}\), then compare with finite-difference approximations.