Interactions Between the Generalized Hadamard Product and the Eigenvalues of Symmetric Matrices
نویسنده
چکیده
In [8] a notion of generalized Hadamard product was introduced. We show that when certain kinds of tensors interact with the eigenvalues of symmetric matrices the resulting formulae can be nicely expressed using the generalized Hadamard product and two simple linear operations on the tensors. The Calculus-type rules developed here will be used in [9] to routinize, to a large extend, the differentiation of spectral functions. We include all the necessary definitions and results from [8] on the generalized Hadamard product to make the reading self-contained.
منابع مشابه
Generalized matrix functions, determinant and permanent
In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the de...
متن کاملA mathematically simple method based on denition for computing eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices
In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to th...
متن کاملWeak log-majorization inequalities of singular values between normal matrices and their absolute values
This paper presents two main results that the singular values of the Hadamard product of normal matrices $A_i$ are weakly log-majorized by the singular values of the Hadamard product of $|A_{i}|$ and the singular values of the sum of normal matrices $A_i$ are weakly log-majorized by the singular values of the sum of $|A_{i}|$. Some applications to these inequalities are also given. In addi...
متن کاملOn the Higher-Order Derivatives of Spectral Functions: Two Special Cases
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 d...
متن کامل