Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex ${\displaystyle n\times n}$ matrix A is the set

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

where ${\displaystyle \mathbf {x} ^{*}}$ denotes the conjugate transpose of the vector ${\displaystyle \mathbf {x} }$. 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.

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

Properties

Let sum of sets denote a sumset.

General properties

1. The numerical range is the range of the Rayleigh quotient.
2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
3. ${\displaystyle W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}}$ for all square matrix ${\displaystyle A}$ and complex numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. Here ${\displaystyle I}$ is the identity matrix.
4. ${\displaystyle W(A)}$ is a subset of the closed right half-plane if and only if ${\displaystyle A+A^{*}}$ is positive semidefinite.
5. The numerical range ${\displaystyle W(\cdot )}$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. ${\displaystyle W(UAU^{*})=W(A)}$ for any unitary ${\displaystyle U}$.
7. ${\displaystyle W(A^{*})=W(A)^{*}}$.
8. If ${\displaystyle A}$ is Hermitian, then ${\displaystyle W(A)}$ is on the real line. If ${\displaystyle A}$ is anti-Hermitian, then ${\displaystyle W(A)}$ is on the imaginary line.
9. ${\displaystyle W(A)=\{z\}}$ if and only if ${\displaystyle A=zI}$.
10. (Sub-additive) ${\displaystyle W(A+B)\subseteq W(A)+W(B)}$.
11. ${\displaystyle W(A)}$ contains all the eigenvalues of ${\displaystyle A}$.
12. The numerical range of a ${\displaystyle 2\times 2}$ matrix is a filled ellipse.
13. ${\displaystyle W(A)}$ is a real line segment ${\displaystyle [\alpha ,\beta ]}$ if and only if ${\displaystyle A}$ is a Hermitian matrix with its smallest and the largest eigenvalues being ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.

Normal matrices

1. 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 ${\textstyle \langle x,Ax\rangle \in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)}$.
2. If ${\displaystyle A}$ is a normal matrix then ${\displaystyle W(A)}$ is the convex hull of its eigenvalues.
3. If ${\displaystyle \alpha }$ is a sharp point on the boundary of ${\displaystyle W(A)}$, then ${\displaystyle \alpha }$ is a normal eigenvalue of ${\displaystyle A}$.

Numerical radius

1. ${\displaystyle r(\cdot )}$ is a unitarily invariant norm on the space of ${\displaystyle n\times n}$ matrices.
2. ${\displaystyle r(A)\leq \|A\|_{op}\leq 2r(A)}$, where ${\displaystyle \|\cdot \|_{op}}$ denotes the operator norm.^[1][2][3][4]
3. ${\displaystyle r(A)=\|A\|_{op}}$ if (but not only if) ${\displaystyle A}$ is normal.
4. ${\displaystyle r(A^{n})\leq r(A)^{n}}$.

Proofs

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.

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 $${\displaystyle {\frac {1}{2}}I+{\frac {1}{2}}{\begin{bmatrix}\cos 2\theta &e^{i\phi }\sin 2\theta \\e^{-i\phi }\sin 2\theta &-\cos 2\theta \end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}}$$ where ${\textstyle x,y,z}$, 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 closed unit sphere, so it is compact.

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

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 $${\displaystyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}\succeq 0}$$ and that inequality is sharp, meaning that ${\textstyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}}$ 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:

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

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

Since the above holds for all ${\textstyle \theta \in [\phi _{1},\phi _{2}]}$, we must have: $${\displaystyle \langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle =0\quad \forall y\in \mathbb {C} ^{n}.}$$

For any ${\textstyle y\in \mathbb {C} ^{n}}$ and ${\textstyle \alpha \in \mathbb {C} }$, substitute ${\textstyle \alpha y}$ into the equation: $${\displaystyle \alpha \langle y,Ax_{0}\rangle +\alpha ^{*}\langle x_{0},Ay\rangle =0.}$$ Choose ${\textstyle \alpha =1}$ and ${\textstyle \alpha =i}$, then simplify, we obtain ${\displaystyle \langle y,Ax_{0}\rangle =0}$ for all ${\displaystyle y}$, 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, $${\displaystyle |\langle v,Av\rangle |\leq \|v\|_{2}\|Av\|_{2}=\|Av\|_{2}\leq \|A\|_{op}}$$

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

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 ${\textstyle \|B\|_{op}=r(B)\leq r(A),\|C\|_{op}=r(iC)\leq r(A)}$.

Generalisations

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

See also

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

Bibliography

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References

1. ""well-known" inequality for numerical radius of an operator". StackExchange.
2. "Upper bound for norm of Hilbert space operator". StackExchange.
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4. Hilary Priestley. "B4b hilbert spaces: extended synopses 9. Spectral theory" (PDF). 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.