English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Perturbation or error bounds of functions have been of great interest for along time. If the functions are differentiable, then the mean value theorem andTaylor's theorem come handy for this purpose. While the former is useful inestimating $\|f(A+X)-f(A)\|$ in terms of $\|X\|$ and requires the norms of thefirst derivative of the function, the latter is useful in computing higherorder perturbation bounds and needs norms of the higher order derivatives ofthe function.  In the study of matrices, determinant is an important function. Other scalarvalued functions like eigenvalues and coefficients of characteristic polynomialare also well studied. Another interesting function of this category is thepermanent, which is an analogue of the determinant in matrix theory. Moregenerally, there are operator valued functions like tensor powers,antisymmetric tensor powers and symmetric tensor powers which have gainedimportance in the past. In this article, we give a survey of the recent work onthe higher order derivatives of these functions and their norms. Using Taylor'stheorem, higher order perturbation bounds are obtained. Some of these resultsare very recent and their detailed proofs will appear elsewhere.

Derivatives of Multilinear Functions of Matrices