On the number of matrices to generate a matrix \(\ast\)-algebra over the real field
Let \(M_n(\mathbb{R})\) be the set of matrices of order \(n\) over the real field \(\mathbb{R}\). A subset \(\mathcal{T}\) of \(M_n(\mathbb{R})\) is said to be a matrix \(*\)-algebra if \(I_n \in \mathcal{T}\) and \(\mathcal{T}\) is closed under linear combination, multiplication, and transposition. In the present paper, among other things, it is shown that any matrix \(*\)-algebra over \(\mathbb{R}\) can be generated with at most four randomly-chosen symmetric matrices. The proof of this characterization result involves a skillful use of a structure theorem for matrix \(*\)-algebras and the notion of genericity in the eigenvalue structure.
scientific article; zbMATH DE number 6125863