Numerical range

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

$W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \not =0\right\}$

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).

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.

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

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. $W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}$ for all square matrix $A$ and complex numbers $\alpha$ and $\beta$. Here $I$ is the identity matrix.
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 $W(\cdot )$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. $W(UAU^{*})$ 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. $W(A)=\{z\}$ if and only if $A=zI$.
10. (Sub-additive) $W(A+B)\subseteq W(A)+W(B)$.
11. $W(A)$ contains all the eigenvalues of $A$.
12. The numerical range of a $2\times 2$ matrix is a filled ellipse.
13. $W(A)$ 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

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 $A$ is a normal matrix then $W(A)$ is the convex hull of its eigenvalues.
3. If $\alpha$ is a sharp point on the boundary of $W(A)$, then $\alpha$ is a normal eigenvalue of $A$.

Numerical radius

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

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 $${\frac {1}{2}}I+{\frac {1}{2}}{\begin{bmatrix}\cos 2\theta &e^{i\phi }\sin 2\theta \\e^{-i\phi }\sin 2\theta &-\cos 2\theta \end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}$$ 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 $$\langle z,Az\rangle =\langle z,P^{*}APz\rangle \in W(P^{*}AP)\subset W(A)$$

Proof of (5)

Let ${\textstyle W}$ satisfy these properties.

By linearity, ${\textstyle W(aI)=\{a\}}$ for any complex ${\textstyle a}$.

Let ${\textstyle A}$ be diagonal and positive semidefinite, then we can multiply ${\textstyle A}$ by any ${\textstyle e^{i\phi }}$ where ${\textstyle \phi \in [-\pi /2,+\pi /2]}$, and obtain some ${\textstyle A'=e^{i\phi }A}$ such that ${\textstyle A+A^{*}\succeq 0}$, so ${\textstyle e^{i\phi }W(A)\subset \mathbb {C} ^{+}}$, meaning that ${\textstyle W(A)}$ must be on the half-line ${\textstyle \mathbb {R} ^{+}}$.

By the “only if” part, ${\textstyle W(A)}$ is the line segment ${\textstyle [a,b]}$ where ${\textstyle a,b}$ are the minimum and maximum of the diagonal entries of ${\textstyle A}$.

Let ${\textstyle A=UDU^{*}}$ be normal. By a similar argument, any ${\textstyle \phi \in \mathbb {R} ,\lambda \in \mathbb {C} }$, we have ${\textstyle e^{i\phi }(W(A)-\lambda )\subset \mathbb {C} ^{+}}$ iff ${\textstyle Re(e^{i\phi }(D_{kk}-\lambda ))\geq 0}$ for each ${\textstyle k}$. Varying ${\textstyle \lambda ,\phi }$ allows arbitrary translation and rotation of the complex plane, so we find that the closure of ${\textstyle W(A)}$ is the convex hull of ${\textstyle D_{11},\dots ,D_{nn}}$. Since ${\textstyle W(A)}$ is also closed, it is equal to this convex hull.

In general, by the same argument by convex hull, $$W(A)=\{z\in \mathbb {C} :\forall t,\phi \in \mathbb {R} ,e^{i\phi }A+e^{-i\phi }A^{*}\succeq tI{\text{ iff }}e^{i\phi }z+e^{-i\phi }z^{*}\geq t\}$$ - Since ${\textstyle e^{i\phi }A+e^{-i\phi }A^{*}\succeq tI}$ is a stable property under perturbation of ${\textstyle A}$, ${\textstyle W(A)}$ is a continuous function of ${\textstyle A}$.

Then, since ${\textstyle W}$ is the numerical range on normal matrices, and the normal matrices are dense in the space of all matrices, ${\textstyle W}$ is the numerical range.

Normal matrices

Proof of (1), (2)

For (2), if ${\textstyle A}$ is normal, then it has a full eigenbasis. Now apply (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:

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

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 ${\textstyle r(B),r(iC)\leq r(A)}$.

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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