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

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

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.

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

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. (Sub-additive) ${\displaystyle W(A+B)\subseteq W(A)+W(B)}$, where the sum on the right-hand side denotes a sumset.
7. ${\displaystyle W(A)}$ contains all the eigenvalues of ${\displaystyle A}$.
8. The numerical range of a ${\displaystyle 2\times 2}$ matrix is a filled ellipse.
9. ${\displaystyle W(A)}$ 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 ${\displaystyle \beta }$.
10. If ${\displaystyle A}$ is a normal matrix then ${\displaystyle W(A)}$ is the convex hull of its eigenvalues.
11. If ${\displaystyle \alpha }$ is a sharp point on the boundary of ${\displaystyle W(A)}$, then ${\displaystyle \alpha }$ is a normal eigenvalue of ${\displaystyle A}$.
12. ${\displaystyle r(\cdot )}$ is a norm on the space of ${\displaystyle n\times n}$ matrices.
13. ${\displaystyle r(A)\leq \|A\|\leq 2r(A)}$, where ${\displaystyle \|\cdot \|}$ denotes the operator norm.^[1][2][3][4]
14. ${\displaystyle r(A^{n})\leq r(A)^{n}}$

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

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

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