Lesson plan
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|.
Article summary
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|.
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