Not every square matrix \(A \in {\mathbb R}^{n \times n}\) is diagonalisable. However, if the characteristic polynomial \(\chi _A\) decomposes into linear factors, at least a Schur decomposition exists (see Chap. 42 ). The Jordan normal form is somewhat an improvement of the Schur decomposition: It exists under the same conditions as the Schur decomposition and is a particularly simple upper triangular matrix: Apart from some ones on the upper sub-diagonal it has a diagonal shape. The essential thing now is that for every complex matrix A such a Jordan normal formJ exists. Determining the matrix S that transforms A into Jordan normal form J, i.e. the matrix S with \(J = S^{-1} A \, S\) , is somewhat laborious: The first step is to determine the generalised eigenspaces. We will do this in the present chapter, in the next chapter we will show how to obtain S from this.