Description

Eigenvalues are often introduced as the main tool for understanding matrices - but they don't tell the whole story. This project asks a more subtle question: How big can a matrix really behave, and how do we measure that size properly? To answer this, we study matrix norms, the so-called spectral radius, and matrix functions like powers and inverses.

The problems guide you from familiar linear algebra facts into surprising phenomena: matrices that behave "larger" than their eigenvalues suggest, and norms that capture hidden growth.

The first term will revolve around learning basic concepts and results on matrix norms, following the book by Horn&Johnson (reference [2] below, mainly Section 5.6). By the end of term we will have learned the following:

• For an \(N\times N\) matrix \(A\) we study the operator norm which depends on the norm \(\|\cdot\|_X\) imposed on the underlying finite-dimensional space \(X=\mathbb{C}^N\) as follows: \[\|A\|:=\sup_{\|x\|_X=1}\|Ax\|_X.\]
• We study the operator norm if \(X\) is equipped with the \(p\)-norm: With the \(1\)-norm we obtain the maximum column sum matrix norm, with the \(\infty\)-norm we get the maximum row sum matrix norm, and with the \(2\)-norm we arrive at the maximal singular value \(s\) of \(A\), that is, \(s^2\) is the maximal eigenvalue of \(A^*A\).
• We study inequalities between \(\|A\|\) and the spectral radius \[r(A):=\max\{|\lambda|:\,\lambda \text{ eigenvalue of } A\}.\] Analogously we find inequalities between \(\|(A-z)^{-1}\|\) and the distance of \(z\) to the spectrum \(\sigma(A)\), the set of eigenvalues. Also, for a polynomial \(p\), we find \[\|p(A)\| \geq \max_{z\in\sigma(A)}|p(z)|.\]
• We prove Gelfand's formula: \[r(A) = \lim_{k\to\infty} \|A^k\|^{1/k}.\]

After laying these foundations together, in the second term you would be free to choose your preferred direction in which to continue your investigation. A few examples of topics you would be able to explore include (but are not limited to):

• The Crouzeix conjecture, which is an unsolved problem in matrix analysis (even open for general \(3\times 3\) matrices). It stipulates that, for any polynomial \(p\), \[\|p(A)\|\leq 2 \max_{z\in W(A)}|p(z)|\] where \(W(A):=\{\langle A x,x\rangle_{2}:\,\|x\|_{2}=1\}\) is the so-called numerical range of \(A\).
• Pseudospectra: For \(\varepsilon>0\) we study \[\sigma_{\varepsilon}(A):=\{z\in\mathbb{C}:\,\|(A-z)^{-1}\|>1/\varepsilon\}\] and prove the surprising connection to eigenvalue perturbations: \[\sigma_{\varepsilon}(A)=\bigcup_{\|B\|<\varepsilon}\sigma(A+B).\]
• The norm of analytic matrix-valued functions \(A(z)\) for \(z\in\mathbb{C}\), and the connection to the maximum principle: The map \(z\mapsto \|A(z)\|\) is subharmonic, for example for \(A(z)=(A-z)^{-1}\). The map can be constant on a subset without necessarily being constant everywhere.

Mode of operation and evidence of learning

This project will revolve around learning through reading with a focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding.

Students will demonstrate their understanding of the subject matter by solving relevant problems, exploring and constructing examples, investigating theoretical applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites and Companion modules

Prerequisites: Analysis I, Linear Algebra I, Complex Analysis II.
Analysis III is recommended but not necessary.

Co-requisites: None.
Several of the results that we will explore in this project in the setting of matrices have analogues in the setting of linear operators in infinite-dimensional spaces. To work in infinite-dimensional spaces, we would appeal to methods of Functional Analysis that you get to explore if you take the module Functional Analysis & Applications IV. It is thus recommended (but not necessary) to take this module alongside the project.

References

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

[2] R.A. Horn and C.R. Johnson. Matrix Analysis (2nd ed.). Cambridge University Press (2012). Available online via Durham library.

[3] https://en.wikipedia.org/wiki/Crouzeix's_conjecture

[4] L.N. Trefethen and M. Embree. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press (2005). Available online via Durham library.