Special families of matrices. Matrices of class δ
Reduction of endomorphisms

The subspace Ker M ² has a dimension that varies between two bounds: the dimension of Ker A and its double. Matrices such as Ker A = Ker A ²

and matrices such as

dim(Ker A ² ) = 2 dim(Ker A)

form two families that deserve special attention. The following propositions give some characteristic properties of one or other of these two families. Keep in mind the exact sequence

$\left\{0\right\}\to \mathrm{Ker}\mathrm{\hspace{0.17em}}A\to \mathrm{Ker}\mathrm{\hspace{0.17em}}{A}^2$