Numerical range

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 \times n matrix A is the set 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 \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). Equivalently, the elements of W(A) are of the form \operatorname{tr}(AP) , where P is a Hermitian projection operator from \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 \big\{ |\lambda| : \lambda \in W(A) \big\} = \sup_{\|x\|_2=1} \big|\langle\mathbf{x}, A\mathbf{x} \rangle\big|.

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 \times n matrix A is the set 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\} 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(\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. 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) = \{z\} 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] 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{span}(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{hull}\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\|_{\operatorname{op}} \leq 2r(A) , where \|\cdot\|_{\operatorname{op}} denotes the operator norm. r(A) = \|A\|_{\operatorname{op}} if (but not only if) A is normal. r(A^n) \le r(A)^n . {{Math proof|title=Proof of (2)|proof= W(A) is the image of a continuous map x \mapsto \langle x,Ax\rangle from the \mathbb{PC}^n , so it is compact. Given two complex nonzero vectors x, y , let P_x, P_y be their corresponding Hermitian projectors from \mathbb{C}^n to their respective spans. Let P be the Hermitian projector to the span of both. We have that P^*AP is an operator on \operatorname{Span}(x, y) . Therefore, the “restricted numerical range” of P^*AP , defined by \{\operatorname{Tr}(P^*APP_z) : z \in \operatorname{Span}(x, y), z \neq 0\} , is a closed ellipse, according to (12). It is also the case that if z \in \operatorname{Span}(x,y) is nonzero, then \operatorname{Tr}(P^*APP_z) = \operatorname{Tr}(APP_zP) = \operatorname{Tr}(AP_z) \in W(A) . Therefore, the restricted numerical range is contained in the full numerical range of A . Thus, if W(A) contains \operatorname{Tr}(AP_x), \operatorname{Tr}(AP_y) , then it contains a closed ellipse that also contains \operatorname{Tr}(AP_x), \operatorname{Tr}(AP_y) , so it contains the line segment between them. }} W(A) = \{\lambda \in \C : PMP = \lambda P \text{ for some Hermitian projector } P \text{ of rank }1\}. This allows a generalization to higher-rank numerical ranges, one for each k = 1, 2, 3, \dots : W_k(A) = \{\lambda \in \C : PMP = \lambda P \text{ for some Hermitian projector } P \text{ of rank }k\}. W_k(A) is always closed and convex, but it might be empty. It is guaranteed to be nonempty if k , and there exists some A such that W_k(A) is empty if k \geq n/3 + 1 . ==See also== Spectral theory Rayleigh quotient Workshop on Numerical Ranges and Numerical Radii ==Bibliography== Books . Papers . . . ==References==

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