In the mathematical field of linear algebra and convex analysis, the numerical range or field of values or Wertvorrat or Wertevorrat of a complex n × n {\displaystyle n\times n} matrix A is the set
W ( A ) = { x ∗ A x x ∗ x | x ∈ C n , x ≠ 0 } = { ⟨ x , A x ⟩ | x ∈ C n , ‖ x ‖ 2 = 1 } {\displaystyle W(A)=\left.\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\;\right|\;\mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \neq 0\right\}={\big \{}\langle \mathbf {x} ,A\mathbf {x} \rangle \;{\big |}\;\mathbf {x} \in \mathbb {C} ^{n},\ \|\mathbf {x} \|_{2}=1{\big \}}}
where x ∗ {\displaystyle \mathbf {x} ^{*}} denotes the conjugate transpose of the vector x {\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). Equivalently, the elements of W ( A ) {\displaystyle W(A)} are of the form tr ( A P ) {\displaystyle \operatorname {tr} (AP)} , where P {\displaystyle P} is a Hermitian projection operator from C n {\displaystyle \mathbb {C} ^{n}} to a one-dimensional subspace. 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 { | λ | : λ ∈ W ( A ) } = sup ‖ x ‖ 2 = 1 | ⟨ x , A x ⟩ | . {\displaystyle r(A)=\sup {\big \{}|\lambda |:\lambda \in W(A){\big \}}=\sup _{\|x\|_{2}=1}{\big |}\langle \mathbf {x} ,A\mathbf {x} \rangle {\big |}.}
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 ( α A + β I ) = α W ( A ) + { β } {\displaystyle W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}} for all square matrix A {\displaystyle A} and complex numbers α {\displaystyle \alpha } and β {\displaystyle \beta } . Here I {\displaystyle I} is the identity matrix.
W ( A ) {\displaystyle W(A)} is a subset of the closed right half-plane if and only if A + A ∗ {\displaystyle A+A^{*}} is positive semidefinite. The numerical range W ( ⋅ ) {\displaystyle W(\cdot )} is the only function on the set of square matrices that satisfies (2), (3) and (4).
W ( U A U ∗ ) = W ( A ) {\displaystyle W(UAU^{*})=W(A)} for any unitary U {\displaystyle U} .
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