Numerical range

Properties

Proofs

General properties

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

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

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

The space of all one-dimensional subspaces of ${\textstyle \mathbb {C} ^{2}}${\textstyle \mathbb {C} ^{2}} is ${\textstyle \mathbb {P} \mathbb {C} ^{1}}${\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 ${\textstyle P}${\textstyle P} are of the form $${\displaystyle {\frac {1}{2}}I+{\frac {1}{2}}{\begin{bmatrix}\cos 2\theta &e^{i\phi }\sin 2\theta \\e^{-i\phi }\sin 2\theta &-\cos 2\theta \end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}}$${\displaystyle {\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 ${\textstyle x,y,z}${\textstyle x,y,z}, satisfying ${\textstyle x^{2}+y^{2}+z^{2}=1}${\textstyle x^{2}+y^{2}+z^{2}=1}, is a point on the unit 2-sphere.

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

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

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

Therefore, the "restricted numerical range" of ${\textstyle P^{*}AP}${\textstyle P^{*}AP}, defined by ${\textstyle \{\operatorname {Tr} (P^{*}APP_{z}):z\in \operatorname {Span} (x,y),z\neq 0\}}${\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 ${\textstyle z\in \operatorname {Span} (x,y)}${\textstyle z\in \operatorname {Span} (x,y)} is nonzero, then ${\textstyle \operatorname {Tr} (P^{*}APP_{z})=\operatorname {Tr} (APP_{z}P)=\operatorname {Tr} (AP_{z})\in W(A)}${\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 ${\textstyle A}${\textstyle A}.

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

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

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

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

The two properties of ${\textstyle W}${\textstyle W} then imply that $${\displaystyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}\succeq 0}$${\displaystyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}\succeq 0} and that inequality is sharp, meaning that ${\textstyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}}${\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 ${\textstyle W(A)}${\textstyle W(A)}.

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

Normal matrices

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

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

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

$${\displaystyle \langle x,Ax\rangle =\sum _{i,j=1}^{k}c_{i}^{*}c_{j}\left\langle v_{i},\lambda _{j}v_{j}\right\rangle =\sum _{i=1}^{k}\left|c_{i}\right|^{2}\lambda _{i}\in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)}$${\displaystyle \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 ${\textstyle W}${\textstyle W}, we can translate and rotate the complex plane, so that we reduce to the case where ${\textstyle \partial W(A)}${\textstyle \partial W(A)} has a sharp point at ${\textstyle 0}${\textstyle 0}, and that the two supporting planes at that point both make an angle ${\textstyle \phi _{1},\phi _{2}}${\textstyle \phi _{1},\phi _{2}} with the imaginary axis, such that ${\textstyle \phi _{1}<\phi _{2},e^{i\phi _{1}}\neq e^{i\phi _{2}}}${\textstyle \phi _{1}<\phi _{2},e^{i\phi _{1}}\neq e^{i\phi _{2}}} since the point is sharp.

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

By general property (4), the numerical range lies in the sectors defined by: $${\displaystyle \operatorname {Re} \left(e^{i\theta }\langle x,Ax\rangle \right)\geq 0\quad {\text{for all }}\theta \in [\phi _{1},\phi _{2}]{\text{ and nonzero }}x\in \mathbb {C} ^{n}.}$${\displaystyle \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 ${\textstyle x=x_{0}}${\textstyle x=x_{0}}, the directional derivative in any direction ${\textstyle y}${\textstyle y} must vanish to maintain non-negativity. Specifically:
$${\displaystyle \left.{\frac {d}{dt}}\operatorname {Re} \left(e^{i\theta }\langle x_{0}+ty,A(x_{0}+ty)\rangle \right)\right|_{t=0}=0\quad \forall y\in \mathbb {C} ^{n},\theta \in [\phi _{1},\phi _{2}].}$${\displaystyle \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:
$${\displaystyle \operatorname {Re} \left(e^{i\theta }\left(\langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle \right)\right)=0\quad \forall y\in \mathbb {C} ^{n},\theta \in [\phi _{1},\phi _{2}].}$${\displaystyle \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 ${\textstyle \theta \in [\phi _{1},\phi _{2}]}${\textstyle \theta \in [\phi _{1},\phi _{2}]}, we must have: $${\displaystyle \langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle =0\quad \forall y\in \mathbb {C} ^{n}.}$${\displaystyle \langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle =0\quad \forall y\in \mathbb {C} ^{n}.}

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

Numerical radius

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

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

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

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

Generalisations

Higher-rank numerical range

See also

Bibliography

References