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

$W(A)=\{\frac{{\mathbf{𝐱}}^{*}A\mathbf{𝐱}}{{\mathbf{𝐱}}^{*}\mathbf{𝐱}}\mid \mathbf{𝐱}\in {\mathbb{ℂ}}^n,\mathbf{𝐱}=0\}=\{⟨\mathbf{𝐱},A\mathbf{𝐱}⟩\mid \mathbf{𝐱}\in {\mathbb{ℂ}}^n,\Vert \mathbf{𝐱}{\Vert }_2=1\}$

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.

$r(A)=\sup \{|\lambda |:\lambda \in W(A)\}=\underset{\Vert x{\Vert }_2=1}{\sup }|⟨\mathbf{𝐱},A\mathbf{𝐱}⟩|.$

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. $W(UA{U}^{*})=W(A)$ 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. $W(A)=\{z\}$ 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 $A$ is normal, and $x\in \mathrm{span}(v_1,\dots ,v_k)$, where $v_1,\dots ,v_k$ are eigenvectors of $A$ corresponding to ${\lambda }_1,\dots ,{\lambda }_k$, respectively, then $⟨x,Ax⟩\in \mathrm{hull}({\lambda }_1,\dots ,{\lambda }_k)$.
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. $r(A)\le \Vert A{\Vert }_{\mathrm{op}}\le 2r(A)$, where $\Vert \cdot {\Vert }_{\mathrm{op}}$ denotes the operator norm.
3. $r(A)=\Vert A{\Vert }_{\mathrm{op}}$ if (but not only if) $A$ is normal.
4. $r(A^n)\le r(A{)}^n$.

Proofs

Most of the claims are obvious. Some are not.

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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