Nilpotent Matrices and the Jordan Canonical Form

David F. Delchamps
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David F. Delchamps: Cornell University, School of Electrical Engineering

Chapter 11 in State Space and Input-Output Linear Systems, 1998, pp 150-161 from Springer

Abstract: Abstract 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.

Date: 1998
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DOI: 10.1007/978-1-4612-3816-4_12

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