Solving the Problem of Simultaneous Diagonalization of Complex Symmetric Matrices via Congruence

Bustamante, Miguel D.; Mellon, Pauline; Velasco, María V.

doi:10.1137/19m1280430

Cited by 9 publications

(19 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Theorem 2 we show that if A is a real algebra and B is a basis of A then B also is a basis of A C , the complexification of A (with the same multiplication structure matrices) and that A is an evolution algebra if, and only if, A C is an evolution algebra and has a natural basis consisting of elements of A. This reduction of the real case to the complex one allows us to apply the results in [25] to both real and complex algebras.…”

Section: Introductionmentioning

confidence: 85%

“…It is worth remarking at this point that the general problem of diagonalisation via congruence considers m symmetric matrices of dimension n × n, where m need not be equal to n. This problem has applications in statistical signal processing and multivariate statistics [21][22][23][24] and was solved for complex symmetric matrices in [25]. Theorem 1.…”

Section: Definitionmentioning

confidence: 99%

“…Since the problem of simultaneous diagonalisation of matrices via congruence was solved in [25] for complex symmetric matrices, we consider the following.…”

Section: Proofmentioning

confidence: 99%

“…In [25], example 16, we give two real matrices which are diagonalisable via congruence by means of a complex matrix but not by means of any real matrix.…”

Section: Proofmentioning

confidence: 99%

“…It has connections with other problems such as blind-source separation in signal processing [21][22][23][24]. The SDC-problem was solved recently for complex symmetric matrices in [25].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Determining When an Algebra Is an Evolution Algebra

2020

Self Cite

View full text Add to dashboard Cite

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n-dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n×n matrices {M1,…,Mn} describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.

show abstract

Section: Introductionmentioning

confidence: 85%

Section: Definitionmentioning

confidence: 99%

“…Since the problem of simultaneous diagonalisation of matrices via congruence was solved in [25] for complex symmetric matrices, we consider the following.…”

Section: Proofmentioning

confidence: 99%

“…In [25], example 16, we give two real matrices which are diagonalisable via congruence by means of a complex matrix but not by means of any real matrix.…”

Section: Proofmentioning

confidence: 99%

“…It has connections with other problems such as blind-source separation in signal processing [21][22][23][24]. The SDC-problem was solved recently for complex symmetric matrices in [25].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations