Semisimple matrix

Let \(k\) be a field with algebraic closure \(k^\alg\). A matrix \(A \in \Mat_{n\times n}(k)\) is semisimple if \(A\) is diagonalizable over \(k^\alg\), that is, if there exist \(P\in\GL_n(k^\alg)\) and a diagonal \(D\in\Mat_{n\times n}(k^\alg)\) with \(A = PDP^{-1}\).