Correction: Bustamante et al. Determining When an Algebra Is an Evolution Algebra. Mathematics 2020, 8, 1349

Miguel D. Bustamante; Pauline Mellon; M. Victoria Velasco

doi:10.3390/math9111289

,

and

¹

School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland

²

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

^*

Author to whom correspondence should be addressed.

Mathematics2021, 9(11), 1289;https://doi.org/10.3390/math9111289

Version Notes

Order Reprints

The authors wish to make the following corrections to this paper [] (see corrected version in postprint []):

On page 2, paragraph 4, complete the first sentence ‘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’ with the phrase ‘and has a natural basis consisting of elements of A’.
Replace Theorem 2 (statement and proof) with

Theorem 2.

Let A be a real algebra. Then 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. Moreover, if A is a real evolution algebra, then every natural basis of A is a natural basis of

A_{ℂ}

.

Proof.

If A is an evolution algebra and if B is a natural basis of A, then obviously B is a natural basis of

A_{ℂ}

. The converse direction is clear.

3.: Replace Corollary 1 with

Corollary 1.

Let A be a real commutative algebra, let B = {e₁, …, e_n} be a basis, and let M₁, …, M_n be the m-structure matrices of A with respect to B. Then, A is an evolution algebra if, and only if, the matrices M₁, …, M_n (regarded as complex matrices) are simultaneously diagonalisable via congruence by means of a real matrix.

4.: Add the following sentence after Corollary 1:

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.

Finally, to aid the reader we note that reference [25] in the corrected paper [] corresponds to reference [] below.

The authors apologize for any inconvenience caused and state that the scientific conclusions are unaffected. The original article has been updated.

Conflicts of Interest

The authors declare no conflict of interest.

References

Bustamante, M.D.; Mellon, P.; Velasco, M.V. Determining When an Algebra Is an Evolution Algebra. Mathematics 2020, 8, 1349. [Google Scholar] [CrossRef]
Bustamante, M.D.; Mellon, P.; Velasco, M.V. Determining When an Algebra Is an Evolution Algebra. arXiv 2021, Postprint. arXiv:2102.04493 [math.RA]. [Google Scholar]
Bustamante, M.D.; Mellon, P.; Velasco, M.V. Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence. SIAM J. Matrix Anal. Appl. 2020, 41, 1616. [Google Scholar] [CrossRef]

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Correction: Bustamante et al. Determining When an Algebra Is an Evolution Algebra. Mathematics 2020, 8, 1349

Conflicts of Interest

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