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Evolution algebras are non-associative algebras that describe non-Mendelianhereditary processes and have connections with many other areas. In this paperwe obtain necessary and sufficient conditions for a given algebra $A$ to be anevolution 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 certainset of $n$ symmetric $n\times n$ matrices $\{M_{1}, \ldots, M_{n}\}$ describingthe product of $A$ are $SDC$. We apply this characterisation to show that whilecertain classical genetic algebras (representing Mendelian and auto-tetraploidinheritance) are not themselves evolution algebras, arbitrarily smallperturbations of these are evolution algebras. This is intriguing as evolutionalgebras model asexual reproduction unlike the classical ones.

Determining when an algebra is an evolution algebra