Project III (MATH3382) 2024-25

The Cauchy integral formula for matrices and applications

Sabine Boegli

Background

Let \(A\) be an \(n\times n\) matrix and \(f:\mathbb C\to\mathbb C\) a holomorphic function. For a polynomial \(f(z)=\sum_{n=0}^N c_n z^n\) we can insert \(f(A)=\sum_{n=0}^N c_n A^n\), but we wonder how to define \(f(A)\) for a general holomorphic function \(f\). Recall that the Cauchy integral formula says \[f(a)=\frac{1}{2\pi i}\int_{\Gamma} \frac{f(z)}{z-a}\,dz\] where \(\Gamma\) is a closed contour with the complex number \(a\) in its interior. This motivates the definition \[f(A):=\frac{1}{2\pi i}\int_{\Gamma} f(z)(z-A)^{-1}\,dz\] where \((z-A)^{-1}\) is the inverse matrix and the closed contour \(\Gamma\) is assumed to have all eigenvalues of \(A\) in its interior. One can show the following properties:

  • For two holomorphic functions \(f,g\) one has \[(f+ g)(A)=f(A)+g(A), \quad (f\cdot g)(A)=f(A)g(A), \quad (f\circ g)(A)=f(g(A)).\]
  • For polynomials \(f\), we recover the obvious definition of \(f(A)\). More generally, for a convergent series \(f(z)=\sum_{n=0}^{\infty}c_n z^n\) we have \(f(A)=\sum_{n=0}^\infty c_n A^n\). This can be used for example to define \(\exp(A)\).
  • Spectral mapping theorem: The eigenvalues of \(f(A)\) are \[\sigma(f(A))=\{f(\lambda):\,\lambda\in\sigma(A)\}\] where \(\sigma(A)\) denotes the set of eigenvalues (called the spectrum) of \(A\).
Instead of \(\Gamma\) enclosing all eigenvalues, one can choose it to enclose only one or a few eigenvalues and study \(\frac{1}{2\pi i}\int_{\Gamma} f(z)(z-A)^{-1}\,dz\) in that case. For example, if \(\Gamma\) encloses one eigenvalue \(\lambda\), then \[\frac{1}{2\pi i}\int_{\Gamma} (z-A)^{-1}\,dz\] is a projection onto the eigenspace corresponding to \(\lambda\). This can be used to study perturbation of eigenvalues and eigenvectors if the matrix \(A\) is perturbed, e.g. by adding a matrix of small norm.

Another variation of the above definition is to allow functions \(f\) that are not holomorphic in all of the complex plane. For example if \(f(z)=z^{1/2}\), we wonder whether \(f(A)\) is well-defined and is a square-root of the matrix \(A\), i.e. whether \(f(A)f(A)=A\).

Description of the project

You will familiarise yourself with the new concept and its properties. For a first impression, see reference [1] below. References [2] and [3] contain systematic treatments of the topic. More generally, \(f(A)\) is defined for a bounded linear operator \(A\) in a Banach space. Then, individually, you will focus on different applications, e.g. convergence of eigenvalues and eigenspaces, or non-entire holomorphic functions \(f\).

Prerequisites

None. It is suggested (but not necessarily required) to take Analysis III alongside the project, as it will help working with Banach spaces.

References

[1] https://en.wikipedia.org/wiki/Holomorphic_functional_calculus

[2] N. Dunford and J.T. Schwartz. Linear Operators, Part I: General Theory. Interscience (1958).

[3] I. Gohberg, S. Goldberg and M. A. Kaashoek. Classes of Linear Operators: Volume 1. Birkhauser (1991).

email: Sabine Boegli