Suppose a matrix $M$ over some field $k$ allows a diagonalisation $P D P^{-1}$. On the level of linear maps, this is a decomposition of the linear map $f: V \to V$ represented by $M$ into
$$V \cong \bigoplus_{i = 1}^n k \xrightarrow{\bigoplus_{i = 1}^n f_i} \bigoplus_{i = 1}^n k \cong V,$$
where $f_i$ is defined by $x \mapsto M_{ii} x$ and the isomorphisms are represented by $P$.
We cannot always do this, so can we find a best approximation in the sense of a decomposition
$$V \cong \bigoplus_{i = 1}^m V_i \xrightarrow{\bigoplus_{i = 1}^m f_i} \bigoplus_{i = 1}^m V_i \cong V,$$
where we want $m$ as large as possible?