Nilpotent Matrices and the Jordan Canonical Form

In this section, we derive the Jordan canonical form for an arbitrary Cn x n ) real or complex matrix A. The Jordan canonical form of A is simply the matrix of the linear transformation $${}_{A}\hat{T}:{{C}^{n}} \to$$ C n with respect to a special basis for C n . We saw in §9 that if the distinct eigenvalues of A are F (λ1),…, λ s , with respective generalized eigenspaces F(λ1),…, F(λ s ), and if z i is a basis for F(λ i ), 1 ⩽ i ⩽ s then the matrix of A with respect to the ordered basis z1 U cial form Uz s for C n takes the special form (*) $$B = \left[ {\begin{array}{*{20}{c}} {{{A}_{1}}} \hfill & 0 \hfill & \cdot \hfill & \cdot \hfill & \cdot \hfill \\ 0 \hfill & {{{A}_{2}}} \hfill & \cdot \hfill & \cdot \hfill & \cdot \hfill \\ \cdot \hfill & 0 \hfill & \cdot \hfill & \cdot \hfill & \cdot \hfill \\ \cdot \hfill & \cdot \hfill & \cdot \hfill & {{{A}_{{s - 1}}}} \hfill & \cdot \hfill \\ \cdot \hfill & \cdot \hfill & \cdot \hfill & 0 \hfill & {{{A}_{s}}} \hfill \\ \end{array} } \right],$$ where each Ai is an (r i × r i ) matrix (r i is the algebraic multiplicity of the eigenvalue λ i ) which satisfies (A i-λ i Ir i ) ri = 0.