numerical range

Let sum of sets denote a sumset.

General properties

1. The numerical range is the range of the Rayleigh quotient.
2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
3. $W(\backslash alpha\; A+\backslash beta\; I)=\backslash alpha\; W(A)+\backslash \{\backslash beta\backslash \}$ for all square matrix $A$ and complex numbers $\backslash alpha$ and $\backslash beta$. Here $I$ is the identity matrix.
4. $W(A)$ is a subset of the closed right half-plane if and only if $A+A^*$ is positive semidefinite.
5. The numerical range $W(\backslash cdot)$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. $W(UAU^*)\; =\; W(A)$ for any unitary $U$.
7. $W(A^*)\; =\; W(A)^*$.
8. 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.
9. $W(A)\; =\; \backslash \{z\backslash \}$

if and only if $A\; =\; zI$.

1. (Sub-additive) $W(A+B)\backslash subseteq\; W(A)+W(B)$.
2. $W(A)$ contains all the eigenvalues of $A$.
3. The numerical range of a $2\; \backslash times\; 2$ matrix is a filled ellipse.
4. $W(A)$ is a real line segment $[\backslash alpha,\; \backslash beta]$ if and only if $A$ is a Hermitian matrix with its smallest and the largest eigenvalues being $\backslash alpha$ and $\backslash beta$.

Normal matrices

1. If $A$ is normal, and $x\; \backslash in\; \backslash operatorname\{span\}(v\_1,\; \backslash dots,\; v\_k)$, where $v\_1,\; \backslash ldots,\; v\_k$ are eigenvectors of $A$ corresponding to $\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_k$, respectively, then $\backslash langle\; x,Ax\backslash rangle\; \backslash in\; \backslash operatorname\{hull\}\backslash left(\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_k\backslash right)$.
2. If $A$ is a normal matrix then $W(A)$ is the convex hull of its eigenvalues.
3. If $\backslash alpha$ is a sharp point on the boundary of $W(A)$, then $\backslash alpha$ is a normal eigenvalue of $A$.

Numerical radius

1. $r(\backslash cdot)$ is a unitarily invariant norm on the space of $n\; \backslash times\; n$ matrices.
2. $r(A)\; \backslash leq\; \backslash |A\backslash |\_\{\backslash operatorname\{op\}\}\; \backslash leq\; 2r(A)$, where $\backslash |\backslash cdot\backslash |\_\{\backslash operatorname\{op\}\}$ denotes the operator norm.{{Cite web | url=https://math.stackexchange.com/questions/3278149/ |title =

"well-known" inequality for numerical radius of an operator | website=StackExchange}}{{Cite web | url=https://math.stackexchange.com/questions/597880/ |title =

Upper bound for norm of Hilbert space operator | website=StackExchange}}{{Cite web |url=https://math.stackexchange.com/questions/4020968/ |title=Inequalities for numerical radius of complex Hilbert space operator |website=StackExchange}}{{Cite web|url=https://web.archive.org/web/20141225190025id_/http://www0.maths.ox.ac.uk:80/system/files/coursematerial/2014/3075/33/14B4b-extsyn9.pdf| title=

B4b hilbert spaces: extended synopses 9. Spectral theory

|author=Hilary Priestley|author-link=Hilary Priestley|quote = In fact, ‖T‖ = max(−m_T , M_T) = w_T. This fails for non-self-adjoint operators, but w_T ≤ ‖T‖ ≤ 2w_T in the complex case.}}

1. $r(A)\; =\; \backslash |A\backslash |\_\{\backslash operatorname\{op\}\}$ if (but not only if) $A$ is normal.
2. $r(A^n)\; \backslash le\; r(A)^n$.

Most of the claims are obvious. Some are not.

{{Math proof|title=Proof of (13)|proof=

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

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

}}

{{Math proof|title=Proof of (12)|proof=

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

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

In more detail, such $P$ are of the form $$$$

\frac 12 I + \frac 12 \begin{bmatrix}\cos2\theta & e^{i\phi} \sin 2\theta \\ e^{-i\phi} \sin 2\theta & -\cos2\theta \end{bmatrix} = \frac 12 \begin{bmatrix}1 + z & x + iy \\ x - iy & 1-z \end{bmatrix}

where $x,\; y,\; z$, satisfying $x^2+y^2+z^2\; =1$, is a point on the unit 2-sphere.

Therefore, the elements of $W(A)$, regarded as elements of $\backslash R^2$ is the composition of two real linear maps and $M\; \backslash mapsto\; \backslash operatorname\{tr\}(AM)$, which maps the 2-sphere to a filled ellipse.

}}

{{Math proof|title=Proof of (2)|proof=

$W(A)$ is the image of a continuous map $x\; \backslash mapsto\; \backslash langle\; x,Ax\backslash rangle$ from the closed unit sphere, so it is compact.

For any $x,\; y$ of unit norm, project $A$ to the span of $x,\; y$ as $P^*AP$. Then $W(P^*AP)$ is a filled ellipse by the previous result, and so for any $\backslash theta\; \backslash in\; [0,1]$, let $z\; =\; \backslash theta\; x\; +\; (1\; -\; \backslash theta)y$, we have $$\backslash langle\; z,\; Az\backslash rangle\; =\; \backslash langle\; z,\; P^*APz\backslash rangle\; \backslash in\; W(P^*AP)\; \backslash subset\; W(A)$$

}}

{{Math proof|title=Proof of (5)|proof=

Let $W$ satisfy these properties. Let $W\_0$ be the original numerical range.

Fix some matrix $A$. We show that the supporting planes of $W(A)$ and $W\_0(A)$ are identical. This would then imply that $W(A)\; =\; W\_0(A)$ since they are both convex and compact.

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

The two properties of $W$ then imply that $$$$

e^{i\phi}(A-z) + e^{-i\phi}(A-z)^* \succeq 0

and that inequality is sharp, meaning that $e^\{i\backslash phi\}(A-z)\; +\; e^\{-i\backslash phi\}(A-z)^*$ has a zero eigenvalue. This is a complete characterization of the supporting planes of $W(A)$.

The same argument applies to $W\_0(A)$, so they have the same supporting planes.

}}

{{Math proof|title=Proof of (1), (2)|proof=

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

Since $A$ is normal, by the spectral theorem, there exists a unitary matrix $U$ such that $A=U\; D\; U^*$, where $D$ is a diagonal matrix containing the eigenvalues $\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots,\; \backslash lambda\_n$ of $A$.

Let $x=c\_1\; v\_1+c\_2\; v\_2+\backslash cdots+c\_k\; v\_k$. Using the linearity of the inner product, that $A\; v\_j=\backslash lambda\_j\; v\_j$, and that $\backslash left\backslash \{v\_i\backslash right\backslash \}$ are orthonormal, we have:

\langle x, A x\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)

}}

{{Math proof|title=Proof (3)|proof=

By affineness of $W$, we can translate and rotate the complex plane, so that we reduce to the case where $\backslash partial\; W(A)$ has a sharp point at $0$, and that the two supporting planes at that point both make an angle $\backslash phi\_1,\; \backslash phi\_2$ with the imaginary axis, such that since the point is sharp.

Since $0\; \backslash in\; W(A)$, there exists a unit vector $x\_0$ such that $x\_0^*\; Ax\_0\; =\; 0$.

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 $x\; =\; x\_0$, the directional derivative in any direction $y$ 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 $\backslash theta\; \backslash in\; [\backslash phi\_1,\; \backslash phi\_2]$, we must have: $$$$

\langle y, Ax_0 \rangle + \langle x_0, Ay \rangle = 0 \quad \forall y \in \mathbb{C}^n.

For any $y\; \backslash in\; \backslash mathbb\{C\}^n$ and $\backslash alpha\; \backslash in\; \backslash mathbb\{C\}$, substitute $\backslash alpha\; y$ into the equation: $$$$

\alpha \langle y, Ax_0 \rangle + \alpha^* \langle x_0, Ay \rangle = 0.

Choose $\backslash alpha\; =\; 1$ and $\backslash alpha\; =\; i$, then simplify, we obtain $\backslash langle\; y,\; Ax\_0\; \backslash rangle\; =\; 0$ for all $y$, thus $Ax\_0\; =\; 0$.

}}

{{Math proof|title=Proof of (2)|proof=

Let $v\; =\; \backslash arg\backslash max\_\{\backslash |x\backslash |\_2=\; 1\}\; |\backslash langle\; x,Ax\backslash rangle|$. We have $r(A)\; =\; |\backslash langle\; v,Av\backslash rangle|$.

By Cauchy–Schwarz, $$$$

|\langle v,Av\rangle| \leq \|v\|_2 \|Av\|_2 = \|Av\|_2 \leq \|A\|_{op}

For the other one, let $A\; =\; B\; +\; iC$, where $B,\; C$ are Hermitian. $$$$

\|A\|_{op} \leq \|B \|_{op} + \|C \|_{op}

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

}}

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

• Spectral theory
• Rayleigh quotient
• Workshop on Numerical Ranges and Numerical Radii

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