In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

and

convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of som ...

, the numerical range or field of values of a

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

n \times n

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

''A'' is the set :

W(A) = \left\

where

\mathbf^*

denotes the

conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...

of the

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

\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

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

s 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 Ax, x \rangle, .

Properties

# The numerical range is the

range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...

of the

Rayleigh quotient In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as: R(M,x) = . For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...

. # (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). # (Sub-additive)

W(A+B)\subseteq W(A)+W(B)

, where the sum on the right-hand side denotes a

sumset In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is, :A + B = \. The n-f ...

. #

W(A)

contains all the

s 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 In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...

with its smallest and the largest eigenvalues being

\alpha

and

\beta

. # If

A

is a

normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...

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

. #

r(\cdot)

is a norm on the space of

n \times n

matrices. #

r(A) \leq \, A\,  \leq 2r(A)

, where

\, \cdot\,

denotes the operator norm. #

r(A^n) \le r(A)^n

Generalisations

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

Bibliography

*. *. *. *. *. * *. *. * "Functional Characterizations of the Field of Values and the Convex Hull of the Spectrum", Charles R. Johnson, ''Proceedings of the American Mathematical Society'', 61(2):201-204, Dec 1976. {{DEFAULTSORT:Numerical Range Matrix theory Spectral theory Operator theory Linear algebra

Properties

Generalisations

See also

Bibliography