The Gaussian Toeplitz matrix
Copyright policy )The Gaussian Toeplitz matrix
Let be \(T_{ij}:=(2\pi\sigma^ 2)^{-1/2}\exp[-(i-j)^ 2/(2\sigma^ 2)]\) for \(i,j=0,1,\dots,N-1\). The authors prove that the matrix \(T:=[T_{ij}]\) is positive definite for all values of \(\sigma>0\) and \(N\geq 1\). Analytic expressions are given for the Cholesky decomposition \(T=LL^ T\), for the determinant of \(T\), and for the inverse \(T^{-1}\). Exact bounds for the eigenvalues of \(T\) are also derived.
Gaussian Toeplitz matrix, Numerical Analysis, Algebra and Number Theory, positive definite, inverse, Determinants, permanents, traces, other special matrix functions, Direct numerical methods for linear systems and matrix inversion, determinant, Inequalities involving eigenvalues and eigenvectors, Factorization of matrices, eigenvalue bounds, Discrete Mathematics and Combinatorics, Centre for Theoretical Studies, Hermitian, skew-Hermitian, and related matrices, Theory of matrix inversion and generalized inverses, Geometry and Topology, Cholesky decomposition, Electrical Engineering
Gaussian Toeplitz matrix, Numerical Analysis, Algebra and Number Theory, positive definite, inverse, Determinants, permanents, traces, other special matrix functions, Direct numerical methods for linear systems and matrix inversion, determinant, Inequalities involving eigenvalues and eigenvectors, Factorization of matrices, eigenvalue bounds, Discrete Mathematics and Combinatorics, Centre for Theoretical Studies, Hermitian, skew-Hermitian, and related matrices, Theory of matrix inversion and generalized inverses, Geometry and Topology, Cholesky decomposition, Electrical Engineering
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