Numerical range

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In the mathematical field of linear algebra and convex analysis, the numerical range of a square matrix with complex entries is a subset of the complex plane associated to the matrix. If A is an n × n matrix with complex entries, then the numerical range of A is the range of the Rayleigh quotient

$f_A(\mathbf{x}) = \frac{\mathbf{x}^*A\mathbf{x}}{\mathbf{x}^*\mathbf{x}}$

where x* denotes the Hermitian adjoint of the vector x. Thus the numerical range of A is the set

$\nu(A) = \left\{\frac{\mathbf{x}^*A\mathbf{x}}{\mathbf{x}^*\mathbf{x}} \mid x\in\mathbb{C}^n\right\}.$

It is easy to show that the points of ν(A) furthest from the origin are eigenvalues of A. Furthermore, the spectrum of A is contained within ν(A). By the Hausdorff-Toeplitz theorem, if A is a normal matrix, then the numerical range is a convex polygon in the complex plane whose vertices are eigenvalues of A.