The numerical range (also known as the field of values) of a square matrix is a fundamental concept in linear algebra that characterizes the set of all possible complex numbers obtained by applying the matrix to unit vectors. It provides valuable insights into the behavior and properties of a matrix, particularly concerning its eigenvalues and diagonal entries.
Understanding the Numerical Range
For a given $n \times n$ complex matrix $A$, its numerical range, often denoted as $W(A)$, is defined as the set of complex numbers:
$$W(A) = {x^Ax : x \in \mathbb{C}^n, x^x = 1}$$
Here, $x$ is a column vector in $\mathbb{C}^n$ (a vector of complex numbers), and $x^*$ denotes its conjugate transpose. This definition means that the numerical range comprises all scalar values that result from the "action" of the matrix $A$ on any unit vector $x$.
The numerical range is always a convex and compact subset of the complex plane. This property, known as the Toeplitz-Hausdorff Theorem, is a cornerstone of its study. Its boundary provides crucial information about the matrix's characteristics.
Inclusions within the Numerical Range
A key aspect of the numerical range is that it encompasses several important properties of the matrix:
- Diagonal Entries: The numerical range explicitly includes the diagonal entries of the matrix. These are obtained when the unit vector $x$ is chosen to be the standard basis vectors (unit vectors along the coordinate axes), for instance, $e_k = [0, \dots, 1, \dots, 0]^T$ where the '1' is in the $k$-th position.
- Eigenvalues: All eigenvalues of the matrix are contained within its numerical range. Eigenvalues are special values $\lambda$ for which $Ax = \lambda x$ for some non-zero vector $x$ (an eigenvector). If $x$ is normalized to be a unit vector, then $x^Ax = x^(\lambda x) = \lambda(x^*x) = \lambda$. This means that the eigenvalues form a subset of the numerical range.
Significance and Applications
The numerical range is a powerful tool in various fields, offering more information than just eigenvalues, especially for non-normal matrices (matrices that do not commute with their conjugate transpose).
- Stability Analysis: In systems theory and control, the numerical range can help analyze the stability of dynamic systems. If the numerical range is confined to the left half of the complex plane, it often indicates stability.
- Numerical Analysis: It provides bounds for iterative methods used to solve linear systems or compute eigenvalues. For example, the smallest eigenvalue of a positive definite matrix can be bounded by the numerical range.
- Operator Theory: In functional analysis, the concept extends to operators on Hilbert spaces, where it plays a crucial role in understanding operator properties.
- Matrix Perturbations: The numerical range can give insights into how the eigenvalues of a matrix might shift under small perturbations.
Understanding the numerical range provides a comprehensive view of a matrix's characteristics, combining geometric insights with algebraic properties.
