Numerical range

{{Short description|Aspect of a numerical matrix}} In the mathematical field of linear algebra and convex analysis, the '''numerical range''' or '''field of values''' or '''Wertvorrat''' or '''Wertevorrat''' of a complex <math>n \times n</math> matrix ''A'' is the set

:<math>W(A) = \left\{\frac{\mathbf{x}^*A\mathbf{x}}{\mathbf{x}^*\mathbf{x}} \mid \mathbf{x}\in\mathbb{C}^n,\ \mathbf{x}\neq 0\right\} = \left\{\langle\mathbf{x}, A\mathbf{x} \rangle \mid \mathbf{x}\in\mathbb{C}^n,\ \|\mathbf{x}\|_2=1\right\}</math>

where <math>\mathbf{x}^*</math> denotes the conjugate transpose of the vector <math>\mathbf{x}</math>. 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 <math display="inline">W(A)</math> are of the form <math display="inline">\operatorname{tr}(AP)</math>, where <math display="inline">P</math> is a Hermitian projection operator from <math display="inline">\C^n</math> 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.

:<math>r(A) = \sup \{ |\lambda| : \lambda \in W(A) \} = \sup_{\|x\|_2=1} |\langle\mathbf{x}, A\mathbf{x} \rangle|.</math>

==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. # <math>W(\alpha A+\beta I)=\alpha W(A)+\{\beta\}</math> for all square matrix <math>A</math> and complex numbers <math>\alpha</math> and <math>\beta</math>. Here <math>I</math> is the identity matrix. # <math>W(A)</math> is a subset of the closed right half-plane if and only if <math>A+A^*</math> is positive semidefinite. # The numerical range <math>W(\cdot)</math> is the only function on the set of square matrices that satisfies (2), (3) and (4). # <math>W(UAU^*) = W(A)</math> for any unitary <math>U</math>. # <math>W(A^*) = W(A)^*</math>. # If <math>A</math> is Hermitian, then <math>W(A)</math> is on the real line. If <math>A</math> is anti-Hermitian, then <math>W(A)</math> is on the imaginary line. # <math>W(A) = \{z\} </math> if and only if <math>A = zI</math>. # (Sub-additive) <math>W(A+B)\subseteq W(A)+W(B)</math>. # <math>W(A)</math> contains all the eigenvalues of <math>A</math>. # The numerical range of a <math>2 \times 2</math> matrix is a filled ellipse. # <math>W(A)</math> is a real line segment <math>[\alpha, \beta]</math> if and only if <math>A</math> is a Hermitian matrix with its smallest and the largest eigenvalues being <math>\alpha</math> and <math>\beta</math>. '''Normal matrices''' # If <math display="inline">A</math> is normal, and <math display="inline">x \in \operatorname{span}(v_1, \dots, v_k)</math>, where <math display="inline">v_1, \ldots, v_k</math> are eigenvectors of <math display="inline">A</math> corresponding to <math display="inline">\lambda_1, \ldots, \lambda_k</math>, respectively, then <math display="inline">\langle x,Ax\rangle \in \operatorname{hull}\left(\lambda_1, \ldots, \lambda_k\right)</math>. # If <math>A</math> is a normal matrix then <math>W(A)</math> is the convex hull of its eigenvalues. # If <math>\alpha</math> is a sharp point on the boundary of <math>W(A)</math>, then <math>\alpha</math> is a normal eigenvalue of <math>A</math>. '''Numerical radius''' # <math>r(\cdot)</math> is a unitarily invariant norm on the space of <math>n \times n</math> matrices. # <math>r(A) \leq \|A\|_{\operatorname{op}} \leq 2r(A) </math>, where <math> \|\cdot\|_{\operatorname{op}}</math> denotes the operator norm.<ref>{{Cite web | url=https://math.stackexchange.com/questions/3278149/ |title = "well-known" inequality for numerical radius of an operator | website=StackExchange}}</ref><ref>{{Cite web | url=https://math.stackexchange.com/questions/597880/ |title = Upper bound for norm of Hilbert space operator | website=StackExchange}}</ref><ref>{{Cite web |url=https://math.stackexchange.com/questions/4020968/ |title=Inequalities for numerical radius of complex Hilbert space operator |website=StackExchange}}</ref><ref>{{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(−mT , MT) = wT. This fails for non-self-adjoint operators, but wT ≤ ‖T‖ ≤ 2wT in the complex case.}}</ref> # <math>r(A) = \|A\|_{\operatorname{op}}</math> if (but not only if) <math>A</math> is normal. # <math>r(A^n) \le r(A)^n</math>.

== Proofs == Most of the claims are obvious. Some are not.

=== General properties === {{Math proof|title=Proof of (13)|proof= If <math display="inline">A</math> is Hermitian, then it is normal, so it is the convex hull of its eigenvalues, which are all real.

Conversely, assume <math display="inline">W(A)</math> is on the real line. Decompose <math display="inline">A = B + C</math>, where <math display="inline">B</math> is a Hermitian matrix, and <math display="inline">C</math> an anti-Hermitian matrix. Since <math display="inline">W(C)</math> is on the imaginary line, if <math display="inline">C \neq 0</math>, then <math display="inline">W(A)</math> would stray from the real line. Thus <math display="inline">C = 0</math>, and <math display="inline">A</math> is Hermitian. }}The following proof is due to<ref>{{Cite journal |last=Davis |first=Chandler |date=June 1971 |title=The Toeplitz-Hausdorff Theorem Explained |url=https://www.cambridge.org/core/product/identifier/S0008439500058197/type/journal_article |journal=Canadian Mathematical Bulletin |language=en |volume=14 |issue=2 |pages=245–246 |doi=10.4153/CMB-1971-042-7 |issn=0008-4395}}</ref>{{Math proof|title=Proof of (12)|proof=

The elements of <math display="inline">W(A)</math> are of the form <math display="inline">\operatorname{tr}(AP)</math>, where <math display="inline">P</math> is projection from <math display="inline">\C^2</math> to a one-dimensional subspace.

The space of all one-dimensional subspaces of <math display="inline">\C^2</math> is <math display="inline">\mathbb P\mathbb C^1</math>, which is a 2-sphere. The image of a 2-sphere under a linear projection is a filled ellipse.

In more detail, such <math display="inline">P</math> are of the form <math display="block"> \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} </math> where <math display="inline">x, y, z</math>, satisfying <math display="inline">x^2+y^2+z^2 =1</math>, is a point on the unit 2-sphere.

Therefore, the elements of <math display="inline">W(A)</math>, regarded as elements of <math display="inline">\R^2</math> is the composition of two real linear maps <math display="inline">(x,y,z) \mapsto \frac 12 \begin{bmatrix}1 + z & x + iy \\ x - iy & 1-z \end{bmatrix}</math> and <math display="inline">M \mapsto \operatorname{tr}(AM)</math>, which maps the 2-sphere to a filled ellipse. }}

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

<math display="inline">W(A)</math> is the image of a continuous map <math display="inline">x \mapsto \langle x,Ax\rangle</math> from the <math>\mathbb{PC}^n</math>, so it is compact.

Given two complex nonzero vectors <math display="inline">x, y</math>, let <math display="inline">P_x, P_y</math> be their corresponding Hermitian projectors from <math display="inline">\mathbb{C}^n</math> to their respective spans. Let <math display="inline">P</math> be the Hermitian projector to the span of both. We have that <math display="inline">P^*AP</math> is an operator on <math display="inline">\operatorname{Span}(x, y)</math>.

Therefore, the “restricted numerical range” of <math display="inline">P^*AP</math>, defined by <math display="inline">\{\operatorname{Tr}(P^*APP_z) : z \in \operatorname{Span}(x, y), z \neq 0\}</math>, is a closed ellipse, according to (12). It is also the case that if <math display="inline">z \in \operatorname{Span}(x,y)</math> is nonzero, then <math display="inline">\operatorname{Tr}(P^*APP_z) = \operatorname{Tr}(APP_zP) = \operatorname{Tr}(AP_z) \in W(A)</math>. Therefore, the restricted numerical range is contained in the full numerical range of <math display="inline">A</math>.

Thus, if <math display="inline">W(A)</math> contains <math display="inline">\operatorname{Tr}(AP_x), \operatorname{Tr}(AP_y)</math>, then it contains a closed ellipse that also contains <math display="inline">\operatorname{Tr}(AP_x), \operatorname{Tr}(AP_y)</math>, so it contains the line segment between them. }}

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

Let <math display="inline">W</math> satisfy these properties. Let <math display="inline">W_0</math> be the original numerical range.

Fix some matrix <math display="inline">A</math>. We show that the supporting planes of <math display="inline">W(A)</math> and <math display="inline">W_0(A)</math> are identical. This would then imply that <math display="inline">W(A) = W_0(A)</math> since they are both convex and compact.

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

The two properties of <math display="inline">W</math> then imply that <math display="block"> e^{i\phi}(A-z) + e^{-i\phi}(A-z)^* \succeq 0 </math> and that inequality is sharp, meaning that <math display="inline">e^{i\phi}(A-z) + e^{-i\phi}(A-z)^*</math> has a zero eigenvalue. This is a complete characterization of the supporting planes of <math display="inline">W(A)</math>.

The same argument applies to <math display="inline">W_0(A)</math>, so they have the same supporting planes.

}}

=== Normal matrices ===

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

For (2), if <math display="inline">A</math> is normal, then it has a full eigenbasis, so it reduces to (1).

Since <math display="inline">A</math> is normal, by the spectral theorem, there exists a unitary matrix <math display="inline">U</math> such that <math display="inline">A=U D U^*</math>, where <math display="inline">D</math> is a diagonal matrix containing the eigenvalues <math display="inline">\lambda_1, \lambda_2, \ldots, \lambda_n</math> of <math display="inline">A</math>.

Let <math display="inline">x=c_1 v_1+c_2 v_2+\cdots+c_k v_k</math>. Using the linearity of the inner product, that <math display="inline">A v_j=\lambda_j v_j</math>, and that <math display="inline">\left\{v_i\right\}</math> are orthonormal, we have:

<math display="block"> \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> }}

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

By affineness of <math display="inline">W</math>, we can translate and rotate the complex plane, so that we reduce to the case where <math display="inline">\partial W(A)</math> has a sharp point at <math display="inline">0</math>, and that the two supporting planes at that point both make an angle <math display="inline">\phi_1, \phi_2</math> with the imaginary axis, such that <math display="inline">\phi_1 < \phi_2, e^{i\phi_1} \neq e^{i\phi_2}</math> since the point is sharp.

Since <math display="inline">0 \in W(A)</math>, there exists a unit vector <math display="inline">x_0</math> such that <math display="inline">x_0^* Ax_0 = 0</math>.

By general property (4), the numerical range lies in the sectors defined by: <math display="block"> \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. </math> At <math display="inline">x = x_0</math>, the directional derivative in any direction <math display="inline">y</math> must vanish to maintain non-negativity. Specifically: <math display="block"> \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]. </math> Expanding this derivative: <math display="block"> \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]. </math>

Since the above holds for all <math display="inline">\theta \in [\phi_1, \phi_2]</math>, we must have: <math display="block"> \langle y, Ax_0 \rangle + \langle x_0, Ay \rangle = 0 \quad \forall y \in \mathbb{C}^n. </math>

For any <math display="inline">y \in \mathbb{C}^n</math> and <math display="inline">\alpha \in \mathbb{C}</math>, substitute <math display="inline">\alpha y</math> into the equation: <math display="block"> \alpha \langle y, Ax_0 \rangle + \alpha^* \langle x_0, Ay \rangle = 0. </math> Choose <math display="inline">\alpha = 1</math> and <math display="inline">\alpha = i</math>, then simplify, we obtain <math>\langle y, Ax_0 \rangle = 0</math> for all <math>y</math>, thus <math display="inline">Ax_0 = 0</math>. }}

=== Numerical radius ===

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

Let <math display="inline">v = \arg\max_{\|x\|_2= 1} |\langle x,Ax\rangle|</math>. We have <math display="inline">r(A) = |\langle v,Av\rangle|</math>.

By Cauchy–Schwarz, <math display="block"> |\langle v,Av\rangle| \leq \|v\|_2 \|Av\|_2 = \|Av\|_2 \leq \|A\|_{op} </math>

For the other one, let <math display="inline">A = B + iC</math>, where <math display="inline">B, C</math> are Hermitian. <math display="block"> \|A\|_{op} \leq \|B \|_{op} + \|C \|_{op} </math>

Since <math display="inline">W(B)</math> is on the real line, and <math display="inline">W(iC)</math> is on the imaginary line, the extremal points of <math display="inline">W(B), W(iC)</math> appear in <math display="inline">W(A)</math>, shifted, thus both <math display="inline">\|B\|_{op} = r(B) \leq r(A), \|C\|_{op} = r(iC) \leq r(A)</math>. }}

== Generalisations == *C-numerical range *Joint numerical range *Product numerical range *Polynomial numerical hull

=== Higher-rank numerical range === The numerical range is equivalent to the following definition:<math display="block">W(A) = \{\lambda \in \C : PMP = \lambda P \text{ for some Hermitian projector } P \text{ of rank }1\}</math>This allows a generalization to '''higher-rank numerical ranges''', one for each <math>k = 1, 2, 3, \dots</math>:<ref>{{Cite journal |last=Choi |first=Man-Duen |last2=Kribs |first2=David W. |last3=Życzkowski |first3=Karol |date=October 2006 |title=Higher-rank numerical ranges and compression problems |url=https://linkinghub.elsevier.com/retrieve/pii/S0024379506001637 |journal=Linear Algebra and its Applications |language=en |volume=418 |issue=2-3 |pages=828–839 |doi=10.1016/j.laa.2006.03.019|url-access=subscription }}</ref><math display="block">W_k(A) = \{\lambda \in \C : PMP = \lambda P \text{ for some Hermitian projector } P \text{ of rank }k\}</math><math>W_k(A)</math> is always closed and convex,<ref>{{Cite journal |last=Li |first=Chi-Kwong |last2=Sze |first2=Nung-Sing |date=2008 |title=Canonical Forms, Higher Rank Numerical Ranges, Totally Isotropic Subspaces, and Matrix Equations |url=https://www.jstor.org/stable/20535511 |journal=Proceedings of the American Mathematical Society |volume=136 |issue=9 |pages=3013–3023 |issn=0002-9939}}</ref><ref>{{Cite journal |last=Woerdeman |first=Hugo J. |date=2008-01-01 |title=The higher rank numerical range is convex |url=https://doi.org/10.1080/03081080701352211 |journal=Linear and Multilinear Algebra |volume=56 |issue=1-2 |pages=65–67 |doi=10.1080/03081080701352211 |issn=0308-1087|url-access=subscription }}</ref> but it might be empty. It is guaranteed to be nonempty if <math>k < n/3+1</math>, and there exists some <math>A</math> such that <math>W_k(A)</math> is empty if <math>k \geq n/3+1</math>.<ref>{{Cite journal |last=Li |first=Chi-Kwong |last2=Poon |first2=Yiu-Tung |last3=Sze |first3=Nung-Sing |date=2009-06-01 |title=Condition for the higher rank numerical range to be non-empty |url=https://doi.org/10.1080/03081080701786384 |journal=Linear and Multilinear Algebra |volume=57 |issue=4 |pages=365–368 |doi=10.1080/03081080701786384 |issn=0308-1087|arxiv=0706.1540 }}</ref>

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==References== {{reflist}}

{{DEFAULTSORT:Numerical Range}} Category:Matrix theory Category:Spectral theory Category:Operator theory Category:Linear algebra