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\ = \left\$

where $\mathbf^*$ denotes the conjugate transpose of the vector $\mathbf$. 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.

:$r(A) = \sup \ = \sup_ , \langle\mathbf, A\mathbf \rangle, .$

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.
# $W(\alpha A+\beta I)=\alpha W(A)+\$ for all square matrix $A$ and complex numbers $\alpha$ and $\beta$. Here $I$ is the identity matrix.
# $W(A)$ is a subset of the closed right half-plane if and only if $A+A^*$ is positive semidefinite.
# The numerical range $W(\cdot)$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
# $W(UAU^*) = W(A)$ for any unitary $U$.
# $W(A^*) = W(A)^*$.
# 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.
# $W(A) = \$ if and only if $A = zI$.
# (Sub-additive) $W(A+B)\subseteq W(A)+W(B)$.
# $W(A)$ contains all the eigenvalues of $A$.
# The numerical range of a $2 \times 2$ matrix is a filled ellipse.
# $W(A)$ is a real line segment alpha, \beta/math> if and only if $A$ is a Hermitian matrix with its smallest and the largest eigenvalues being $\alpha$ and $\beta$.
Normal matrices
# If $A$ is normal, and $x \in \operatorname(v_1, \dots, v_k)$, where $v_1, \ldots, v_k$ are eigenvectors of $A$ corresponding to $\lambda_1, \ldots, \lambda_k$, respectively, then $\langle x,Ax\rangle \in \operatorname\left(\lambda_1, \ldots, \lambda_k\right)$.
# If $A$ is a normal matrix then $W(A)$ is the convex hull of its eigenvalues.
# If $\alpha$ is a sharp point on the boundary of $W(A)$, then $\alpha$ is a normal eigenvalue of $A$.
Numerical radius
# $r(\cdot)$ is a unitarily invariant norm on the space of $n \times n$ matrices.
# $r(A) \leq \, A\, _ \leq 2r(A)$, where $\, \cdot\, _$ denotes the operator norm.
# $r(A) = \, A\, _$ if (but not only if) $A$ is normal.
# $r(A^n) \le r(A)^n$.

Proofs

Most of the claims are obvious. Some are not.

General properties

Normal matrices

Numerical radius

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

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