2010/07/31 by Frank Nielsen, Priyanka Grover, Rajendra Bhatia · 3 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebra over a field #Differentiable function #Eigenvalues and eigenvectors #Geometry #Mathematical Inequalities and Applications #Mathematical analysis #Mathematical proof #Mathematics #Matrix Theory and Algorithms #Matrix function #Perturbation (astronomy) #Pure mathematics #Scalar (mathematics) #Symmetric matrix #Taylor series #Taylor's theorem #math.FA #math.GM #msc:15A15 #msc:15A69
paper · pdf · doi:10.1007/978-3-642-30232-9
published as Matrix Information Geometry (2013) eds. F. Nielsen, R. Bhatia, 93-109 · 17 pages
openalex publication_date 2012/08/03 · crossref created 2012/08/03 · crossref issued 2013/01/01 · crossref published 2013/01/01 · crossref published-print 2013/01/01 · openalex created_date 2016/06/24 · arxiv created 2017/04/01 · arxiv updated 2017/04/04 · crossref deposited 2023/05/11 · crossref indexed 2026/08/03 · openalex updated_date 2026/08/05
Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in estimating ‖f(A+X)-f(A)‖ in terms of ‖X‖ and requires the norms of the first derivative of the function, the latter is useful in computing higher order perturbation bounds and needs norms of the higher order derivatives of the function. In the study of matrices, determinant is an important function. Other scalar valued functions like eigenvalues and coefficients of characteristic polynomial are also well studied. Another interesting function of this category is the permanent, which is an analogue of the determinant in matrix theory. More generally, there are operator valued functions like tensor powers, antisymmetric tensor powers and symmetric tensor powers which have gained importance in the past. In this article, we give a survey of the recent work on the higher order derivatives of these functions and their norms. Using Taylor's theorem, higher order perturbation bounds are obtained. Some of these results are very recent and their detailed proofs will appear elsewhere.