A proof of Sn-i (a11,..., ann) ≥ Sn-i (λ1,..., λn), 0 ≤ i ≤ n - 1 for an n × n positive definite symmetric matrix A = [aij]nxn, using mixed determinants
| dc.contributor.author | Pranayanuntana, Poramate | |
| dc.date.accessioned | 2026-08-06T09:55:10Z | |
| dc.date.available | 2026-08-06T09:55:10Z | |
| dc.date.issued | 2007-01-01 | |
| dc.description.abstract | By using mixed determinant of special form D (A, n - i; I, i) for positive definite symmetric matrix A, in particular the operator concavity property of the map f : A → D<sup>1/(n-i)</sup> (A, n - i; I, i)I together with unital positive linear map Φ : A → A ο I the Hadamard product of A with the identity matrix I, inequalities of the form S<inf>n-i</inf>(a<inf>11</inf>,..., a<inf>nn</inf>) ≥ S<inf>n-i</inf>(λ<inf>1</inf>,..., λ<inf>n</inf>), 0 ≤ i ≤ n - 1 for an n × n positive definite symmetric matrix A = [a<inf>ij</inf>] with S<inf>k</inf> : IR<sup>n</sup> → IR, k = 1, 2,...,n defined by S<inf>k</inf>(χ) := ∑<inf>1</inf>≤i<inf>1</inf><i<inf>2</inf><...< i<inf>k</inf>≤n χ<inf>i1</inf> χ<inf>i2</inf> · χ<inf>ik</inf>, and called the elementary symmetric polynomials and λ<inf>i</inf>, i = 1, 2,...,n the eigenvalues of A, are derived. This result was first proved using Schur-concavity property of the elementary symmetric function S<inf>k</inf> (χ) together with the fact that for any positive definite symmetric matrix A, the vector of its diagonal entries is majorized by the vector of its eigenvalues. Hence, this suggests relationship among mixed determinant, majorization and Schur-concavity. | |
| dc.identifier.citation | Wseas Transactions on Mathematics, 6(1), 195-204, 2007 | |
| dc.identifier.issn | 11092769 | |
| dc.identifier.other | 2-s2.0-33751576834 | |
| dc.identifier.uri | https://dspace.kmitl.ac.th/handle/123456789/1595 | |
| dc.source | Wseas Transactions on Mathematics | |
| dc.subject | Aleksandrov inequality | |
| dc.subject | Doubly stochastic matrix | |
| dc.subject | Elementary symmetric polynomial | |
| dc.subject | Majorization | |
| dc.subject | Matrix Hadamard product | |
| dc.subject | Mixed determinant | |
| dc.subject | Operator concave function | |
| dc.subject | Operator convex function | |
| dc.subject | Operator monotone function | |
| dc.subject | Schur-concave function | |
| dc.subject | Schur-convex function | |
| dc.subject | Unital positive linear map | |
| dc.title | A proof of Sn-i (a11,..., ann) ≥ Sn-i (λ1,..., λn), 0 ≤ i ≤ n - 1 for an n × n positive definite symmetric matrix A = [aij]nxn, using mixed determinants | |
| dc.type | Article |