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Subsets of a matrix algebra over a field that are invariant under conjugationand contain the linear span of each two of their commuting elements aredescribed. They obviously include the subsets of diagonalizable and nilpotentmatrices. In the paper, the case of an algebraically closed field isconsidered. The problem is easily reduced to description of subsets ofdiagonalizable matrices and subsets of nilpotent matrices with the givenproperties. So, among diagonalizable matrices, there are four of such subsets.As for the nilpotent case, it is proved that the subset should be defined bythe condition that the sizes of all Jordan cells of the matrix belong to acertain number set. An explicit criterion is obtained in terms of this set.

On matrix sets invariant under conjugation and taking linear combinations of commuting elements