A positive-definite matrix which is not extendible It has been noted that not all positive-semidefinite matrices are extendible in the multidimensional case. However, no example of such a matrix has been pr... SW Lang - 《IEEE Transactions on Acoustics Speech & Signal Processing》 被引量: 3...
The purpose of this paper is to study the existence and uniqueness of a positive definite solution to the nonlinear matrix equation X = Q − A∗X −1A + B∗X −1B, which is a special stochastic rational Riccati equation arising in stochastic control theory. Methods Ou...
Then there exists a unique infinite invertible upper triangular operator R: \ell ^0 \rightarrow \ell ^0 such that: 1. \textbf{P}(x) = \textbf{Q}(x) R and, 2. RX_P = X_QR. Proof The proof is in two parts. 1. Since both \textbf{P}(x) and \textbf{Q}(x) are ...
A self-adjoint operator has a Hermitian matrix representation in which the Hermitian matrix is equal to the complex conjugate of its transpose in which the matrix rows and columns are exchanged. The Hermitian matrix representation of A^, the conjugate transpose, is (90)A^+=A^T∗. The matri...
If a matrix is not decomposable, it is called indecomposable. If a matrix is decomposable then the corresponding minimal realization is also decomposable. The following can be proved: Let A be a complex n× n matrix. If after reordering the indices, one can decompose A into a direct sum ...
Mahalanobis distances, checking if the covariance is a positive definite matrix. Modified Cholesky factorization for symmetric but not necessarily positive definite matrices. Omnibus test for univariate normality (Jarque-Bera, Doornik-Hansen, Adjusted Lagrange multiplier test and robust version by Gel and...
You may not use the built-in chol() and within your function, although you can use them to validate your answers. Your code does need to confirm that the input matrix M is square and positive definite. Reply Joachim May 18, 2022 5:36 am Hi Ben, Could you share the code you have...
In this case, the Gramian matrix M is a convolution operator, having some sequence a as kernel. Explicit localization estimates for spline-type spaces The N x N Gramian matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to plays a crucial role in the recovery of missing samples; its...
ForMatrixLHSType='symmetric','positive_definite', or'upper_triangular'the upper triangular part of the inputMatrixLHSmust contain the relevant information of the matrix. The strictly lower triangular part of the matrix is not referenced. ForMatrixLHSType='lower_triangular'the lower triangular part ...
Transformation of a matrix into another similar matrix that is diagonal Positive definite matrix A full-rank matrix whose eigenvalues are all strictly positive Normal matrix A matrix that commutes with its conjugate transpose and is unitarily diagonalizable Singular...