Positive definiteness

Linear Algebra Problems

Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018

The concept of positive-definiteness of matrices is established upon symmetrical matrices. Before introducing the concept, the leading principal minors of a given matrix are defined. Assume that a symmetrical matrix A is

Linear Classification

View Chapter

Purchase Book

Published in A. C. Faul, A Concise Introduction to Machine Learning, 2019

A. C. Faul

The covariance matrix is in general positive definitepositive definite, since v^TΣiv^=1Ni∑v∈Civ^T(v−μi)(v−μi)Tv^=1Ni∑v∈Ci[(v−μi)Tv^]2≥0. It is unlikely that the sum of squares equals zero, since this means that all samples shifted by the mean lie in a space orthogonal to v^. In this case, one of the dimensions of the feature space is redundant, and dimensionality reduction should be performed first. Positive definiteness implies that the matrix is non-singular and can be inverted, a property which we will use later.

Online monitoring of autocorrelated multivariate linear profiles via multivariate mixed models

View Article

Journal Information

Published in Quality Technology & Quantitative Management, 2022

Somayeh Khalili, Rassoul Noorossana

We apply the proposed methods to the model described above to monitor profiles in phase II. Checking random effects covariance matrix, , revealed that it is not a positive definite matrix because of a negative eigenvalue. Using some software functions such as ‘Cholesky Factorization’ function in MATLAB, one can simply check positive definiteness condition.

Positive definiteness

Explore chapters and articles related to this topic

Linear Algebra Problems

Linear Classification

Online monitoring of autocorrelated multivariate linear profiles via multivariate mixed models