# 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
    $W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \not =0\right\}=\left\{\langle \mathbf {x} ,A\mathbf {x} \rangle \mid \mathbf {x} \in \mathbb {C} ^{n},\ \|\mathbf {x} \|_{2}=1\right\}$
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.
    $r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{\|x\|_{2}=1}|\langle \mathbf {x} ,A\mathbf {x} \rangle |.$

## Properties

Let sum of sets denote a sumset.
General 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).
* $W(UAU^{*})=W(A)$ for any unitary $U$.
* $W(A^{*})=W(A)^{*}$.
* If $A$ is Hermitian, then $W(A)$ is on the real line. If $A$ is anti-Hermitian, then $W(A)$ is on the imaginary line.
* $W(A)=\{z\}$ if and only if $A=zI$.
* (Sub-additive) $W(A+B)\subseteq W(A)+W(B)$.
* $W(A)$ 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 $\beta$.
Normal matrices
* If A{\textstyle A} is normal, and x∈span⁡(v$_1$, …, v$_k$){\textstyle x\in \operatorname {span} (v_{1}, \dots , v_{k})}, where v$_1$, …, v$_k${\textstyle v_{1}, \ldots , v_{k}} are eigenvectors of A{\textstyle A} corresponding to λ$_1$, …, λ$_k${\textstyle \lambda _{1}, \ldots , \lambda _{k}}, respectively, then ⟨x, Ax⟩∈hull⁡(λ$_1$, …, λ$_k$){\textstyle \langle x, Ax\rangle \in \operatorname {hull} \left(\lambda _{1}, \ldots , \lambda _{k}\right)}.
* 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 $A$.
Numerical radius
* $r(\cdot )$ is a unitarily invariant norm on the space of $n\times n$ matrices.
* $r(A)\leq \|A\|_{\operatorname {op} }\leq 2r(A)$, where $\|\cdot \|_{\operatorname {op} }$ denotes the operator norm.
* $r(A)=\|A\|_{\operatorname {op} }$ if (but not only if) $A$ is normal.
* $r(A^{n})\leq r(A)^{n}$.

## Proofs

Most of the claims are obvious. Some are not.

### General properties

If A{\textstyle A} is Hermitian, then it is normal, so it is the convex hull of its eigenvalues, which are all real.
Conversely, assume W(A){\textstyle W(A)} is on the real line. Decompose A=B+C{\textstyle A=B+C}, where B{\textstyle B} is a Hermitian matrix, and C{\textstyle C} an anti-Hermitian matrix. Since W(C){\textstyle W(C)} is on the imaginary line, if C≠0{\textstyle C\neq 0}, then W(A){\textstyle W(A)} would stray from the real line. Thus C=0{\textstyle C=0}, and A{\textstyle A} is Hermitian.
The elements of W(A){\textstyle W(A)} are of the form tr⁡(AP){\textstyle \operatorname {tr} (AP)}, where P{\textstyle P} is projection from C$^2${\textstyle \mathbb {C} ^{2}} to a one-dimensional subspace.
The space of all one-dimensional subspaces of C$^2${\textstyle \mathbb {C} ^{2}} is PC$^1${\textstyle \mathbb {P} \mathbb {C} ^{1}}, which is a 2-sphere. The image of a 2-sphere under a linear projection is a filled ellipse.
In more detail, such P{\textstyle P} are of the form ${\frac {1}{2}}I+{\frac {1}{2}}{\begin{bmatrix}\cos 2\theta &e^{i\phi }\sin 2\theta \\e^{-i\phi }\sin 2\theta &-\cos 2\theta \end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}$ where x, y, z{\textstyle x, y, z}, satisfying x$^2$+y$^2$+z$^2$=1{\textstyle x^{2}+y^{2}+z^{2}=1}, is a point on the unit 2-sphere.
Therefore, the elements of W(A){\textstyle W(A)}, regarded as elements of R$^2${\textstyle \mathbb {R} ^{2}} is the composition of two real linear maps (x, y, z)↦ 1/2 [1+zx+iyx−iy1−z]{\textstyle (x, y, z)\mapsto {\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}} and M↦tr⁡(AM){\textstyle M\mapsto \operatorname {tr} (AM)}, which maps the 2-sphere to a filled ellipse.
W(A){\textstyle W(A)} is the image of a continuous map x↦⟨x, Ax⟩{\textstyle x\mapsto \langle x, Ax\rangle } from the closed unit sphere, so it is compact.
For any x, y{\textstyle x, y} of unit norm, project A{\textstyle A} to the span of x, y{\textstyle x, y} as P$^∗$AP{\textstyle P^{*}AP}. Then W(P$^∗$AP){\textstyle W(P^{*}AP)} is a filled ellipse by the previous result, and so for any θ∈[0, 1]{\textstyle \theta \in [0, 1]}, let z=θx+(1−θ)y{\textstyle z=\theta x+(1-\theta )y}, we have $\langle z,Az\rangle =\langle z,P^{*}APz\rangle \in W(P^{*}AP)\subset W(A)$
Let W{\textstyle W} satisfy these properties. Let W$_0${\textstyle W_{0}} be the original numerical range.
Fix some matrix A{\textstyle A}. We show that the supporting planes of W(A){\textstyle W(A)} and W$_0$(A){\textstyle W_{0}(A)} are identical. This would then imply that W(A)=W$_0$(A){\textstyle W(A)=W_{0}(A)} since they are both convex and compact.
By property (4), W(A){\textstyle W(A)} is nonempty. Let z{\textstyle z} be a point on the boundary of W(A){\textstyle W(A)}, then we can translate and rotate the complex plane so that the point translates to the origin, and the region W(A){\textstyle W(A)} falls entirely within C$^+${\textstyle \mathbb {C} ^{+}}. That is, for some ϕ∈R{\textstyle \phi \in \mathbb {R} }, the set e$^{iϕ}$(W(A)−z){\textstyle e^{i\phi }(W(A)-z)} lies entirely within C$^+${\textstyle \mathbb {C} ^{+}}, while for any t>0{\textstyle t>0}, the set e$^{iϕ}$(W(A)−z)−tI{\textstyle e^{i\phi }(W(A)-z)-tI} does not lie entirely in C$^+${\textstyle \mathbb {C} ^{+}}.
The two properties of W{\textstyle W} then imply that $e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}\succeq 0$ and that inequality is sharp, meaning that e$^{iϕ}$(A−z)+e$^{−iϕ}$(A−z)$^∗${\textstyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}} has a zero eigenvalue. This is a complete characterization of the supporting planes of W(A){\textstyle W(A)}.
The same argument applies to W$_0$(A){\textstyle W_{0}(A)}, so they have the same supporting planes.

### Normal matrices

For (2), if A{\textstyle A} is normal, then it has a full eigenbasis, so it reduces to (1).
Since A{\textstyle A} is normal, by the spectral theorem, there exists a unitary matrix U{\textstyle U} such that A=UDU$^∗${\textstyle A=UDU^{*}}, where D{\textstyle D} is a diagonal matrix containing the eigenvalues λ$_1$, λ$_2$, …, λ$_n${\textstyle \lambda _{1}, \lambda _{2}, \ldots , \lambda _{n}} of A{\textstyle A}.
Let x=c$_1$v$_1$+c$_2$v$_2$+⋯+c$_k$v$_k${\textstyle x=c_{1}v_{1}+c_{2}v_{2}+\cdots +c_{k}v_{k}}. Using the linearity of the inner product, that Av$_j$=λ$_j$v$_j${\textstyle Av_{j}=\lambda _{j}v_{j}}, and that {v$_i$}{\textstyle \left\{v_{i}\right\}} are orthonormal, we have:
$\langle x,Ax\rangle =\sum _{i,j=1}^{k}c_{i}^{*}c_{j}\left\langle v_{i},\lambda _{j}v_{j}\right\rangle \sum _{i=1}^{k}\left|c_{i}\right|^{2}\lambda _{i}\in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)$
By affineness of W{\textstyle W}, we can translate and rotate the complex plane, so that we reduce to the case where ∂W(A){\textstyle \partial W(A)} has a sharp point at 0{\textstyle 0}, and that the two supporting planes at that point both make an angle ϕ$_1$, ϕ$_2${\textstyle \phi _{1}, \phi _{2}} with the imaginary axis, such that ϕ$_1$<ϕ$_2$, e$^{iϕ1}$≠e$^{iϕ2}${\textstyle \phi _{1}<\phi _{2}, e^{i\phi _{1}}\neq e^{i\phi _{2}}} since the point is sharp.
Since 0∈W(A){\textstyle 0\in W(A)}, there exists a unit vector x$_0${\textstyle x_{0}} such that x$_0$$^∗$Ax$_0$=0{\textstyle x_{0}^{*}Ax_{0}=0}.
By general property (4), the numerical range lies in the sectors defined by: $\operatorname {Re} \left(e^{i\theta }\langle x,Ax\rangle \right)\geq 0\quad {\text{for all }}\theta \in [\phi _{1},\phi _{2}]{\text{ and nonzero }}x\in \mathbb {C} ^{n}.$ At x=x$_0${\textstyle x=x_{0}}, the directional derivative in any direction y{\textstyle y} must vanish to maintain non-negativity. Specifically:
$\left.{\frac {d}{dt}}\operatorname {Re} \left(e^{i\theta }\langle x_{0}+ty,A(x_{0}+ty)\rangle \right)\right|_{t=0}=0\quad \forall y\in \mathbb {C} ^{n},\theta \in [\phi _{1},\phi _{2}].$ Expanding this derivative:
$\operatorname {Re} \left(e^{i\theta }\left(\langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle \right)\right)=0\quad \forall y\in \mathbb {C} ^{n},\theta \in [\phi _{1},\phi _{2}].$
Since the above holds for all θ∈[ϕ$_1$, ϕ$_2$]{\textstyle \theta \in [\phi _{1}, \phi _{2}]}, we must have: $\langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle =0\quad \forall y\in \mathbb {C} ^{n}.$
For any y∈C$^n${\textstyle y\in \mathbb {C} ^{n}} and α∈C{\textstyle \alpha \in \mathbb {C} }, substitute αy{\textstyle \alpha y} into the equation: $\alpha \langle y,Ax_{0}\rangle +\alpha ^{*}\langle x_{0},Ay\rangle =0.$ Choose α=1{\textstyle \alpha =1} and α=i{\textstyle \alpha =i}, then simplify, we obtain $\langle y,Ax_{0}\rangle =0$ for all $y$, thus Ax$_0$=0{\textstyle Ax_{0}=0}.

### Numerical radius

Let v=arg⁡max$_{‖x‖2=1}$|⟨x, Ax⟩|{\textstyle v=\arg \max _{\|x\|_{2}=1}|\langle x, Ax\rangle |}. We have r(A)=|⟨v, Av⟩|{\textstyle r(A)=|\langle v, Av\rangle |}.
By Cauchy–Schwarz, $|\langle v,Av\rangle |\leq \|v\|_{2}\|Av\|_{2}=\|Av\|_{2}\leq \|A\|_{op}$
For the other one, let A=B+iC{\textstyle A=B+iC}, where B, C{\textstyle B, C} are Hermitian. $\|A\|_{op}\leq \|B\|_{op}+\|C\|_{op}$
Since W(B){\textstyle W(B)} is on the real line, and W(iC){\textstyle W(iC)} is on the imaginary line, the extremal points of W(B), W(iC){\textstyle W(B), W(iC)} appear in W(A){\textstyle W(A)}, shifted, thus both ‖B‖$_{op}$=r(B)≤r(A), ‖C‖$_{op}$=r(iC)≤r(A){\textstyle \|B\|_{op}=r(B)\leq r(A), \|C\|_{op}=r(iC)\leq r(A)}.

## Generalisations

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