2004/04/19 by Hristo S. Sendov, Sendov, Hristo S. · 1 citation
Computer Science · Mathematics · #15A69 #47A75 #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical methods in inverse problems #Optimization and Control (math.OC) #Spectral Theory (math.SP) #math.OC #math.SP #msc:15A18 #msc:15A69 #msc:47A75 #msc:49R50 #primary: 49R50 #secondary: 15A18
paper · pdf · doi:10.48550/arxiv.math/0404349
27 pages
arxiv created 2004/04/19 · openalex publication_date 2004/04/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we use the tensorial language developed in [8] and [9] to differentiate functions of eigenvalues of symmetric matrices. We describe the formulae for the k-th derivative of such functions in two cases. The first case concerns the derivatives of the composition of an arbitrary differentiable function with the eigenvalues at a matrix with distinct eigenvalues. The second development describes the derivatives of the composition of a separable symmetric function with the eigenvalues at an arbitrary symmetric matrix. In the concluding section we re-derive the formula for the Hessian of a general spectral function at an arbitrary point. Our approach leads to a shorter, streamlined derivation than the original in [6]. The language we use, based on the generalized Hadamard product, allows us to view the differentiation of spectral functions as a routine calculus-type procedure.