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Numerical range

Aspect of a numerical matrix

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Source updated August 14, 2026

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values or Wertvorrat or Wertevorrat of a complex $n\times n$ matrix A is the set $W(A)=\left.\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\;\right|\;\mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \neq 0\right\}={\big \{}\langle \mathbf {x} ,A\mathbf {x} \rangle \;{\big |}\;\mathbf {x} \in \mathbb {C} ^{n},\ \|\mathbf {x} \|_{2}=1{\big \}}$ 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).

Equivalently, the elements of $W(A)$ are of the form $\operatorname {tr} (AP)$ , where $P$ is a Hermitian projection operator from $\mathbb {C} ^{n}$ to a one-dimensional subspace.

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. $r(A)=\sup {\big \{}|\lambda |:\lambda \in W(A){\big \}}=\sup _{\|x\|_{2}=1}{\big |}\langle \mathbf {x} ,A\mathbf {x} \rangle {\big |}.$

01Properties

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 ${\textstyle A$ is normal, and ${\textstyle x\in \operatorname {span} (v_{1},\dots ,v_{k})$ , where ${\textstyle v_{1},\ldots ,v_{k}$ are eigenvectors of ${\textstyle A$ corresponding to ${\textstyle \lambda _{1},\ldots ,\lambda _{k}$ , respectively, then ${\textstyle \langle x,Ax\rangle \in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)$ .
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.
$r(A^{n})\leq r(A)^{n}$ .

02Proofs

Most of the claims are obvious. Some are not.

General properties

Proof of (13)

If ${\textstyle A$ is Hermitian, then it is normal, so it is the convex hull of its eigenvalues, which are all real.

Conversely, assume ${\textstyle W(A)$ is on the real line. Decompose ${\textstyle A=B+C$ , where ${\textstyle B$ is a Hermitian matrix, and ${\textstyle C$ an anti-Hermitian matrix. Since ${\textstyle W(C)$ is on the imaginary line, if ${\textstyle C\neq 0$ , then ${\textstyle W(A)$ would stray from the real line. Thus ${\textstyle C=0$ , and ${\textstyle A$ is Hermitian.

The following proof is due to

Proof of (12)

The elements of ${\textstyle W(A)$ are of the form ${\textstyle \operatorname {tr} (AP)$ , where ${\textstyle P$ is projection from ${\textstyle \mathbb {C} ^{2}$ to a one-dimensional subspace.

The space of all one-dimensional subspaces of ${\textstyle \mathbb {C} ^{2}$ is ${\textstyle \mathbb {P} \mathbb {C} ^{1}$ , which is a 2-sphere. The image of a 2-sphere under a linear projection is a filled ellipse.

In more detail, such ${\textstyle P$ 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 ${\textstyle x,y,z$ , satisfying ${\textstyle x^{2}+y^{2}+z^{2}=1$ , is a point on the unit 2-sphere.

Therefore, the elements of ${\textstyle W(A)$ , regarded as elements of ${\textstyle \mathbb {R} ^{2}$ is the composition of two real linear maps ${\textstyle (x,y,z)\mapsto {\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}$ and ${\textstyle M\mapsto \operatorname {tr} (AM)$ , which maps the 2-sphere to a filled ellipse.

Proof of (2)

${\textstyle W(A)$ is the image of a continuous map ${\textstyle x\mapsto \langle x,Ax\rangle$ from the $\mathbb {PC} ^{n}$ , so it is compact.

Given two complex nonzero vectors ${\textstyle x,y$ , let ${\textstyle P_{x},P_{y}$ be their corresponding Hermitian projectors from ${\textstyle \mathbb {C} ^{n}$ to their respective spans. Let ${\textstyle P$ be the Hermitian projector to the span of both. We have that ${\textstyle P^{*}AP$ is an operator on ${\textstyle \operatorname {Span} (x,y)$ .

Therefore, the “restricted numerical range” of ${\textstyle P^{*}AP$ , defined by ${\textstyle \{\operatorname {Tr} (P^{*}APP_{z}):z\in \operatorname {Span} (x,y),z\neq 0\}$ , is a closed ellipse, according to (12). It is also the case that if ${\textstyle z\in \operatorname {Span} (x,y)$ is nonzero, then ${\textstyle \operatorname {Tr} (P^{*}APP_{z})=\operatorname {Tr} (APP_{z}P)=\operatorname {Tr} (AP_{z})\in W(A)$ . Therefore, the restricted numerical range is contained in the full numerical range of ${\textstyle A$ .

Thus, if ${\textstyle W(A)$ contains ${\textstyle \operatorname {Tr} (AP_{x}),\operatorname {Tr} (AP_{y})$ , then it contains a closed ellipse that also contains ${\textstyle \operatorname {Tr} (AP_{x}),\operatorname {Tr} (AP_{y})$ , so it contains the line segment between them.

Proof of (5)

Let ${\textstyle W$ satisfy these properties. Let ${\textstyle W_{0}$ be the original numerical range.

Fix some matrix ${\textstyle A$ . We show that the supporting planes of ${\textstyle W(A)$ and ${\textstyle W_{0}(A)$ are identical. This would then imply that ${\textstyle W(A)=W_{0}(A)$ since they are both convex and compact.

By property (4), ${\textstyle W(A)$ is nonempty. Let ${\textstyle z$ be a point on the boundary of ${\textstyle W(A)$ , then we can translate and rotate the complex plane so that the point translates to the origin, and the region ${\textstyle W(A)$ falls entirely within ${\textstyle \mathbb {C} ^{+}$ . That is, for some ${\textstyle \phi \in \mathbb {R}$ , the set ${\textstyle e^{i\phi }(W(A)-z)$ lies entirely within ${\textstyle \mathbb {C} ^{+}$ , while for any ${\textstyle t>0$ , the set ${\textstyle e^{i\phi }(W(A)-z)-tI$ does not lie entirely in ${\textstyle \mathbb {C} ^{+}$ .

The two properties of ${\textstyle W$ then imply that $e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}\succeq 0$ and that inequality is sharp, meaning that ${\textstyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}$ has a zero eigenvalue. This is a complete characterization of the supporting planes of ${\textstyle W(A)$ .

The same argument applies to ${\textstyle W_{0}(A)$ , so they have the same supporting planes.

Normal matrices

Proof of (1), (2)

For (2), if ${\textstyle A$ is normal, then it has a full eigenbasis, so it reduces to (1).

Since ${\textstyle A$ is normal, by the spectral theorem, there exists a unitary matrix ${\textstyle U$ such that ${\textstyle A=UDU^{*}$ , where ${\textstyle D$ is a diagonal matrix containing the eigenvalues ${\textstyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}$ of ${\textstyle A$ .

Let ${\textstyle x=c_{1}v_{1}+c_{2}v_{2}+\cdots +c_{k}v_{k}$ . Using the linearity of the inner product, that ${\textstyle Av_{j}=\lambda _{j}v_{j}$ , and that ${\textstyle \left\{v_{i}\right\}$ are orthonormal, we have:

$\langle x,Ax\rangle =\sum _{i,j=1}^{k}c_{i}^{*}c_{j}\left\langle v_{i},\lambda _{j}v_{j}\right\rangle =\sum _{i=1}^{k}\left|c_{i}\right|^{2}\lambda _{i}\in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)$

Proof (3)

By affineness of ${\textstyle W$ , we can translate and rotate the complex plane, so that we reduce to the case where ${\textstyle \partial W(A)$ has a sharp point at ${\textstyle 0$ , and that the two supporting planes at that point both make an angle ${\textstyle \phi _{1},\phi _{2}$ with the imaginary axis, such that ${\textstyle \phi _{1}<\phi _{2},e^{i\phi _{1}}\neq e^{i\phi _{2}}$ since the point is sharp.

Since ${\textstyle 0\in W(A)$ , there exists a unit vector ${\textstyle x_{0}$ such that ${\textstyle x_{0}^{*}Ax_{0}=0$ .

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 ${\textstyle x=x_{0}$ , the directional derivative in any direction ${\textstyle y$ 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 ${\textstyle \theta \in [\phi _{1},\phi _{2}]$ , we must have: $\langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle =0\quad \forall y\in \mathbb {C} ^{n}.$

For any ${\textstyle y\in \mathbb {C} ^{n}$ and ${\textstyle \alpha \in \mathbb {C}$ , substitute ${\textstyle \alpha y$ into the equation: $\alpha \langle y,Ax_{0}\rangle +\alpha ^{*}\langle x_{0},Ay\rangle =0.$ Choose ${\textstyle \alpha =1$ and ${\textstyle \alpha =i$ , then simplify, we obtain $\langle y,Ax_{0}\rangle =0$ for all $y$ , thus ${\textstyle Ax_{0}=0$ .

Numerical radius

Proof of (2)

Let ${\textstyle v=\arg \max _{\|x\|_{2}=1}|\langle x,Ax\rangle |$ . We have ${\textstyle r(A)=|\langle v,Av\rangle |$ .

By Cauchy-Schwarz, $|\langle v,Av\rangle |\leq \|v\|_{2}\|Av\|_{2}=\|Av\|_{2}\leq \|A\|_{op}$

For the other one, let ${\textstyle A=B+iC$ , where ${\textstyle B,C$ are Hermitian. $\|A\|_{op}\leq \|B\|_{op}+\|C\|_{op}$

Since ${\textstyle W(B)$ is on the real line, and ${\textstyle W(iC)$ is on the imaginary line, the extremal points of ${\textstyle W(B),W(iC)$ appear in ${\textstyle W(A)$ , shifted, thus both ${\textstyle \|B\|_{op}=r(B)\leq r(A),\|C\|_{op}=r(iC)\leq r(A)$ .

03Generalisations

Higher-rank numerical range

The numerical range is equivalent to the following definition: $W(A)=\{\lambda \in \mathbb {C} :PMP=\lambda P{\text{ for some Hermitian projector }}P{\text{ of rank }}1\}.$ This allows a generalization to higher-rank numerical ranges, one for each $k=1,2,3,\dots$ : $W_{k}(A)=\{\lambda \in \mathbb {C} :PMP=\lambda P{\text{ for some Hermitian projector }}P{\text{ of rank }}k\}.$ $W_{k}(A)$ is always closed and convex, but it might be empty. It is guaranteed to be nonempty if $k<n/3+1$ , and there exists some $A$ such that $W_{k}(A)$ is empty if $k\geq n/3+1$ .

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Sources and credits

This article is adapted from the Wikipedia article “Numerical range”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Numerical range

01Properties

02Proofs

General properties

Normal matrices

Numerical radius

03Generalisations

Higher-rank numerical range

Sources and credits

Related topics

Spectral theory

Rayleigh quotient

Workshop on Numerical Ranges and Numerical Radii