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

The authors wish to make the following corrections to this paper [1] (see corrected version in postprint [2]):

• 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 [1] corresponds to reference [3] below.

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