Is there any lower bound known for the minimal number of generators needed to generate the full matrix algebra of real $n\times n$ matrices — when using only symmetric matrices for the generators?
Analogous question for complex matrices — when using only Hermitian matrices for the generators.
I am aware that $3$ generators suffice when using only idempotent generators. This is a result of Naum Krupnik (Minimal number of idempotent generators of matrix algebras over arbitrary field, Comm. Algebra 20 (1992), no. 11, 3251–3257). (Tandfonline link, restricted access)
I am not familiar with this type of results, so this might be well known or easy. Thanks for any tips.