Numerical range
# Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex $n\times n$ matrix A is the set

    $W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \not =0\right\}$

where $\mathbf {x} ^{*}$ denotes the conjugate transpose of the vector $\mathbf {x}$. The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing x equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing x equal to the eigenvectors).

In engineering, numerical ranges are used as a rough estimate of eigenvalues of A. Recently, generalizations of the numerical range are used to study quantum computing.

A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e.

    $r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{\|x\|=1}|\langle Ax,x\rangle |.$

## Properties

* The numerical range is the range of the Rayleigh quotient.
* (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
* $W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}$ for all square matrix $A$ and complex numbers $\alpha$ and $\beta$. Here $I$ is the identity matrix.
* $W(A)$ is a subset of the closed right half-plane if and only if $A+A^{*}$ is positive semidefinite.
* The numerical range $W(\cdot )$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
* (Sub-additive) $W(A+B)\subseteq W(A)+W(B)$, where the sum on the right-hand side denotes a sumset.
* $W(A)$ contains all the eigenvalues of $A$.
* The numerical range of a $2\times 2$ matrix is a filled ellipse.
* $W(A)$ is a real line segment $[\alpha ,\beta ]$ if and only if $A$ is a Hermitian matrix with its smallest and the largest eigenvalues being $\alpha$ and $\beta$.
* If $A$ is a normal matrix then $W(A)$ is the convex hull of its eigenvalues.
* If $\alpha$ is a sharp point on the boundary of $W(A)$, then $\alpha$ is a normal eigenvalue of $A$.
* $r(\cdot )$ is a norm on the space of $n\times n$ matrices.
* $r(A)\leq \|A\|\leq 2r(A)$, where $\|\cdot \|$ denotes the operator norm.
* $r(A^{n})\leq r(A)^{n}$

## Generalisations

* C-numerical range
* Higher-rank numerical range
* Joint numerical range
* Product numerical range
* Polynomial numerical hull
