Linear Transformations on Matrices: the Invariance of the Third Elementary Symmetric Function

1970 ◽  
Vol 22 (4) ◽  
pp. 746-752 ◽  
Author(s):  
Leroy B. Beasley
Keyword(s):  
Linear Transformation ◽  
Symmetric Function ◽  
Similarity Transformation ◽  
Elementary Symmetric Function ◽  
Complex Numbers ◽  
Linear Transformations ◽  
The Third ◽  
Image Position

Let T be a linear transformation on Mn the set of all n × n matrices over the field of complex numbers, . Let A ∈ Mn have eigenvalues λ1, …, λn and let Er(A) denote the rth elementary symmetric function of the eigenvalues of A :Equivalently, Er(A) is the sum of all the principal r × r subdeterminants of A. T is said to preserve Er if Er[T(A)] = Er(A) for all A ∈ Mn. Marcus and Purves [3, Theorem 3.1] showed that for r ≧ 4, if T preserves Er then T is essentially a similarity transformation; that is, either T: A → UAV for all A ∈ Mn or T: A → UAtV for all A ∈ Mn, where UV = eiθIn, rθ ≡ 0 (mod 2π).

Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions

1959 ◽  
Vol 11 ◽  
pp. 383-396 ◽  
Author(s):  
Marvin Marcus ◽  
Roger Purves
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Similarity Transformation ◽  
Elementary Symmetric Function ◽  
Linear Transformations ◽  
Elementary Symmetric Functions ◽  
Image Position

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn let Er(A) be the rth elementary symmetric function of the eigenvalues of A. Our main result states that if 4 ≤ r ≤ n — 1 and Er(T(A)) = Er(A) for A ∈ Mn then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation:No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.

Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

1967 ◽  
Vol 19 ◽  
pp. 281-290 ◽  
Author(s):  
E. P. Botta
Keyword(s):  
Vector Space ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
Linear Maps ◽  
Generalized Matrix ◽  
Image Position

Let Mm(F) be the vector space of m-square matriceswhere F is a field; let f be a function on Mm(F) to some set R. It is of interest to determine the linear maps T: Mm(F) → Mm(F) which 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. Other classical invariants that may be taken for f are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. Dieudonné (1), Hua (2), Jacobs (3), Marcus (4, 6, 8), Mori ta (9), and Moyls (6) have conducted extensive research in this area. A class of matrix functions that have recently aroused considerable interest (4; 7) is the generalized matrix functions in the sense of I. Schur (10).

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.

scholarly journals Linear Transformations on Algebras of Matrices

1959 ◽  
Vol 11 ◽  
pp. 61-66 ◽  
Author(s):  
Marvin Marcus ◽  
B. N. Moyls
Keyword(s):  
Linear Transformation ◽  
Complex Numbers ◽  
List Type ◽  
Hermitian Matrices ◽  
Linear Transformations ◽  
Unimodular Group

Let Mn denote the algebra of n-square matrices over the complex numbers; and let Un, Hn, and Rk denote respectively the unimodular group, the set of Hermitian matrices, and the set of matrices of rank k, in Mn. Let ev(A) be the set of n eigenvalues of A counting multiplicities. We consider the problem of determining the structure of any linear transformation (l.t.) T of Mn into Mn having one or more of the following properties:(a)T(Rk) ⊆ for k = 1, …, n.(b)T(Un) ⊆ Un(c)det T(A) = det A for all A ∈ Hn.(d)ev(T(A)) = ev(A) for all A ∈ Hn.We remark that we are not in general assuming that T is a multiplicative homomorphism; more precisely, T is a mapping of Mn into itself, satisfyingT(aA + bB) = aT(A) + bT(B)for all A, B in Mn and all complex numbers a, b.

An Inequality for Elementary Symmetric Functions

1972 ◽  
Vol 15 (1) ◽  
pp. 133-135 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Geometric Mean ◽  
Binomial Coefficient ◽  
Elementary Symmetric Function ◽  
Elementary Symmetric Functions ◽  
Image Position

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by1E0 = 1 and Er=0(r>m).We define the rth symmetric mean by2where denote the binomial coefficient. If α1 α2,…,αm are positive reals thenwe have two well-known inequalities3and4In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

A Note an a Group Defined by a Quadratic Form

1960 ◽  
Vol 3 (2) ◽  
pp. 143-148 ◽  
Author(s):  
Marvin Marcus ◽  
Nisar A. Khan
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Complex Numbers ◽  
Linear Transformations ◽  
Linear Combinations ◽  
Real Linear ◽  
Image Position

In a recent series of papers [3,4,5], H. Zassenhaus considered the structure of those linear transformations T on real 4-space, R4, into itself that preserve the quadratic form . That is,1.1Define a function ϕ on R4 to the space M 2 of 2-square matrices over the complex numbers as follows:1.2Let G2 be the vector space of matrices generated by all real linear combinations of1.3

An Inequality for Positive Semidefinite Hermitian Matrices(1)

1974 ◽  
Vol 17 (1) ◽  
pp. 131-132
Author(s):  
Russell Merris
Keyword(s):  
Symmetric Group ◽  
Symmetric Function ◽  
Positive Semidefinite ◽  
Elementary Symmetric Function ◽  
Hermitian Matrices ◽  
Image Position

Let A and B be positive semidefinite Hermitian n-square matrices. If A—B is positive semidefinite, write A≥B. Haynsworth [1] has proved that if A≥B then det(A+B)≥det A+n det B.Let G be a subgroup of the symmetric group, Sn, and let λ be a character on G. Letwhere A = (aij) and Er is the rth elementary symmetric function.

Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

1977 ◽  
Vol 29 (5) ◽  
pp. 937-946
Author(s):  
Hock Ong
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Transformation ◽  
Matrix Function ◽  
Multiplicative Group ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
The Matrix ◽  
Image Position

Let F be a field, F* be its multiplicative group and Mn(F) be the vector space of all n-square matrices over F. Let Sn be the symmetric group acting on the set {1, 2, … , n}. If G is a subgroup of Sn and λ is a function on G with values in F, then the matrix function associated with G and X, denoted by Gλ, is defined byand letℐ(G, λ) = { T : T is a linear transformation of Mn(F) to itself and Gλ(T(X)) = Gλ(X) for all X}.

scholarly journals Interconversion of Plasma Free Thyroxine Values from Assay Platforms with Different Reference Intervals Using Linear Transformation Methods

Biology ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 45
Author(s):  
Fanwen Meng ◽  
Jacqueline Jonklaas ◽  
Melvin Khee-Shing Leow
Keyword(s):  
Linear Transformation ◽  
Piecewise Linear ◽  
Equilibrium Dialysis ◽  
Accurate Method ◽  
Reference Intervals ◽  
Thyroid Stimulating Hormone ◽  
Free Thyroxine ◽  
Thyroid Function Tests ◽  
Linear Transformations ◽  
Transformation Methods

Clinicians often encounter thyroid function tests (TFT) comprising serum/plasma free thyroxine (FT4) and thyroid stimulating hormone (TSH) measured using different assay platforms during the course of follow-up evaluations which complicates reliable comparison and interpretation of TFT changes. Although interconversion between concentration units is straightforward, the validity of interconversion of FT4/TSH values from one assay platform to another with different reference intervals remains questionable. This study aims to establish an accurate and reliable methodology of interconverting FT4 by any laboratory to an equivalent FT4 value scaled to a reference range of interest via linear transformation methods. As a proof-of-concept, FT4 was simultaneously assayed by direct analog immunoassay, tandem mass spectrometry and equilibrium dialysis. Both linear and piecewise linear transformations proved relatively accurate for FT4 inter-scale conversion. Linear transformation performs better when FT4 are converted from a more accurate to a less accurate assay platform. The converse is true, whereby piecewise linear transformation is superior to linear transformation when converting values from a less accurate method to a more robust assay platform. Such transformations can potentially apply to other biochemical analytes scale conversions, including TSH. This aids interpretation of TFT trends while monitoring the treatment of patients with thyroid disorders.

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