3. Schur-convex functions and elementary symmetric function inequalities

2019 ◽  
pp. 135-178
Keyword(s):  
Convex Functions ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Schur Convex

Inequalities for Generalized Symmetric Functions

1969 ◽  
Vol 12 (5) ◽  
pp. 615-623 ◽  
Author(s):  
K.V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Generating Series ◽  
Image Position ◽  
Complete Symmetric Function

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined byrespectively.

Linear Transformation on Matrices: The Invariance of a Class of General Matrix Functions. II

1968 ◽  
Vol 20 ◽  
pp. 739-748 ◽  
Author(s):  
Peter Botta
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Matrix Functions ◽  
General Matrix ◽  
Linear Maps

Let Mm(F) be the vector space of m-square matrices X — (Xij), i,j= 1, … , m over a field ƒ;ƒ a function on Mm(F) to some set R. It is of interest to determine the structure of the linear maps T: Mm(F) → Mm(F) that preserve the values of the function ƒ (i.e., ƒ(T(x)) — ƒ(x) for all X). For example, if we take ƒ(x) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank (6). Other classical invariants that may be taken for ƒ are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues.

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