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

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

Generalisations

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