scholarly journals An analog of Thompson's triangle inequality in Euclidean Jordan algebras

2021 ◽  
Vol 37 (37) ◽  
pp. 156-159
Author(s):  
Jiyuan Tao
Keyword(s):  
Linear Algebra ◽  
Jordan Algebra ◽  
Triangle Inequality ◽  
Short Note ◽  
Direct Proof ◽  
Case Analysis ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Euclidean Jordan Algebras

In a recent paper [Linear Algebra Appl., 461:92--122, 2014], Tao et al. proved an analog of Thompson's triangle inequality for a simple Euclidean Jordan algebra by using a case-by-case analysis. In this short note, we provide a direct proof that is valid on any Euclidean Jordan algebras.

A REPRESENTATION THEOREM FOR LYAPUNOV-LIKE TRANSFORMATIONS ON EUCLIDEAN JORDAN ALGEBRAS

2013 ◽  
Vol 15 (04) ◽  
pp. 1340034 ◽  
Author(s):  
JIYUAN TAO ◽  
M. SEETHARAMA GOWDA
Keyword(s):  
Jordan Algebra ◽  
Linear Transformation ◽  
Representation Theorem ◽  
Elementary Proof ◽  
Euclidean Jordan Algebra ◽  
Symmetric Cone ◽  
Jordan Algebras ◽  
Euclidean Jordan Algebras

A Lyapunov-like (linear) transformation L on a Euclidean Jordan algebra V is defined by the condition [Formula: see text]where K is the symmetric cone of V. In this paper, we give an elementary proof (avoiding Lie algebraic ideas and results) of the fact that Lyapunov-like transformations on V are of the form La + D, where a ∈ V, D is a derivation, and La(x) = a ◦ x for all x ∈ V.

scholarly journals Symmetric geodesics on conformal compactifications of Euclidean Jordan algebras

1999 ◽  
Vol 59 (2) ◽  
pp. 187-201
Author(s):  
Sang Youl Lee ◽  
Yongdo Lim ◽  
Chan-Young Park
Keyword(s):  
Jordan Algebra ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Closed Geodesics ◽  
Shilov Boundary ◽  
Torus Knots ◽  
Euclidean Jordan Algebras ◽  
Real Matrices

In this article we define symmetric geodesies on conformal compactifications of Euclidean Jordan algebras and classify symmetric geodesics for the Euclidean Jordan algebra of all n × n symmetric real matrices. Furthermore, we show that the closed geodesics for the Euclidean Jordan algebra of all 2 × 2 symmetric real matrices are realised as the torus knots in the Shilov boundary of a Lie ball.

On Homomorphic Images of Special Jordan Algebras

1954 ◽  
Vol 6 ◽  
pp. 253-264 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Special Jordan Algebra ◽  
Image Position

A linear algebra is called a Jordan algebra if it satisfies the identities(1) ab = ba, (a2b) a = a2(ba).It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that1,where the dot denotes the multiplication in A. Such an algebra S is called a special Jordan algebra.

A NEW POLYNOMIAL INTERIOR-POINT ALGORITHM FOR THE MONOTONE LINEAR COMPLEMENTARITY PROBLEM OVER SYMMETRIC CONES WITH FULL NT-STEPS

2012 ◽  
Vol 29 (02) ◽  
pp. 1250015 ◽  
Author(s):  
G. Q. WANG
Keyword(s):  
Linear Complementarity Problem ◽  
Interior Point ◽  
Complementarity Problem ◽  
Euclidean Jordan Algebra ◽  
Linear Complementarity ◽  
Jordan Algebras ◽  
Symmetric Cones ◽  
Interior Point Algorithm ◽  
Euclidean Jordan Algebras ◽  
Iteration Bound

In this paper, we present a new polynomial interior-point algorithm for the monotone linear complementarity problem over symmetric cones by employing the framework of Euclidean Jordan algebras. At each iteration, we use only full Nesterov and Todd steps. The currently best known iteration bound for small-update method, namely, [Formula: see text], is obtained, where r denotes the rank of the associated Euclidean Jordan algebra and ε the desired accuracy.

scholarly journals Euclidean Jordan algebras and some conditions over the spectra of a strongly regular graph

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 21
Author(s):  
Luís Vieira
Keyword(s):  
Regular Graph ◽  
Strongly Regular Graph ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Inner Product ◽  
Symmetric Matrices ◽  
Identity Matrix ◽  
Jordan Product ◽  
Euclidean Jordan Algebras ◽  
Strongly Regular

Let G be a primitive strongly regular graph G such that the regularity is less than half of the order of G and A its matrix of adjacency, and let 𝒜 be the real Euclidean Jordan algebra of real symmetric matrices of order n spanned by the identity matrix of order n and the natural powers of A with the usual Jordan product of two symmetric matrices of order n and with the inner product of two matrices being the trace of their Jordan product. Next the spectra of two Hadamard series of 𝒜 associated to A2 is analysed to establish some conditions over the spectra and over the parameters of G.

scholarly journals Some inertia theorems in Euclidean Jordan algebras

2009 ◽  
Vol 430 (8-9) ◽  
pp. 1992-2011 ◽  
Author(s):  
M. Seetharama Gowda ◽  
Jiyuan Tao ◽  
Melania Moldovan
Keyword(s):  
Jordan Algebras ◽  
Euclidean Jordan Algebras

SYMMETRIES OF CERTAIN PHYSICAL THEORIES AND THE JORDAN ALGEBRAS

1992 ◽  
Vol 07 (15) ◽  
pp. 3623-3637 ◽  
Author(s):  
R. FOOT ◽  
G. C. JOSHI
Keyword(s):  
Jordan Algebra ◽  
Space Time ◽  
Jordan Algebras ◽  
Bosonic String ◽  
The Other ◽  
Structure Group ◽  
Division Algebras ◽  
Hermitian Matrices ◽  
Higher Dimensional ◽  
Algebraic Framework

It is shown that the sequence of Jordan algebras [Formula: see text], whose elements are the 3 × 3 Hermitian matrices over the division algebras ℝ, [Formula: see text], ℚ and [Formula: see text], can be associated with the bosonic string as well as the superstring. The construction reveals that the space–time symmetries of the first-quantized bosonic string and superstring actions can be related. The bosonic string and the superstring are associated with the exceptional Jordan algebra while the other Jordan algebras in the [Formula: see text] sequence can be related to parastring theories. We then proceed to further investigate a connection between the symmetries of supersymmetric Lagrangians and the transformations associated with the structure group of [Formula: see text]. The N = 1 on-shell supersymmetric Lagrangians in 3, 4 and 6-dimensions with a spin 0 field and a spin 1/2 field are incorporated within the Jordan-algebraic framework. We also make some remarks concerning a possible role for the division algebras in the construction of higher-dimensional extended objects.

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