Open AccessOn the estimation of ${x}^TA^{-1}{x}$ for symmetric matrices
Author(s) -
Paraskevi Fika,
Marilena Mitrouli,
Ondrej Turec
Publication year - 2021
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2021.5611
Subject(s) - mathematics , positive definite matrix , matrix (chemical analysis) , quadratic equation , combinatorics , inverse , symmetric matrix , work (physics) , estimation , eigenvalues and eigenvectors , geometry , physics , management , economics , materials science , quantum mechanics , composite material , thermodynamics
The central mathematical problem studied in this work is the estimation of the quadratic form $x^TA^{-1}x$ for a given symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ and vector $x \in \mathbb{R}^n$. Several methods to estimate $x^TA^{-1}x$ without computing the matrix inverse are proposed. The precision of the estimates is analyzed both analytically and numerically.