Background

The numerical range of an $N\times N$ matrix $A$ is the subset of the complex numbers $\mathbb C$ defined by $$W(A):=\left\{\frac{x^*Ax}{x^*x}:\,x\in\mathbb C^N,\,x\neq 0\right\}$$ where $x^*$ denotes the conjugate transpose of a vector $x\in\mathbb C^N$ (seen as $N\times 1$ matrix). If $A$ is Hermitian, then the numerical range is the (real) interval whose endpoints are the minimal and maximal eigenvalues of $A$. For a more general class of matrices (including all Hermitian ones), the numerical range is a polygonal set with eigenvalues as corners (i.e. $W(A)$ is equal to the convex hull of the eigenvalues). For general matrices $A$, the numerical range has the following properties:

• $W(A)$ contains all eigenvalues of $A$.
• For a $2\times 2$ matrix $A$ the numerical range is an ellipse.
• Toeplitz-Hausdorff theorem: $W(A)$ is a convex subset of $\mathbb C$.

Here you see two examples of numerical ranges. On the left is the numerical range of a diagonal matrix with diagonal entries $-1, 1\pm {\rm i}, 2\pm {\rm i}$, each of these five points being an eigenvalue and a corner of the numerical range. On the right is the numerical range of a "general" matrix whose numerical range has no corners.

Description of the project

You will familiarise yourself with the concept of numerical range, see reference [1] below. Using Chapter 1 of [2] you will learn about the above general properties of $W(A)$. Then, individually, you will focus on different aspects, e.g. corners of the numerical range; numerical implementations to compute the numerical range; or block numerical ranges (a generalisation where the matrix $A$ is first divided into blocks).

Prerequisites

None. If you are interested in numerical implementations, it might be helpful to have some prior knowledge in any programming language/software.

References

[1] https://en.m.wikipedia.org/wiki/Numerical_range

[2] K.E. Gustafson and D.K.M. Rao. Numerical Range: The Field of Values of Linear Operators and Matrices. Springer New York (1995). Also available on google books.