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Mathematics-Online course: Linear Algebra - Normal Forms - Jordan Normal Form

Jordan Form

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A square matrix $ A$ can be brought to block diagonal form by a similarity transformation

$\displaystyle J = \left(\begin{array}{ccc} J_1 & & 0 \\ & \ddots & \\ 0 & & J_k \end{array}\right) = Q^{-1} A Q\,. $

Here the Jordan blocks have the form

$\displaystyle J_i = \left(\begin{array}{ccccc} \lambda_i & 1 & & & 0 \\ 0 & \... ...ts & \\ & & & \lambda_i & 1 \\ 0 & & & & \lambda_i \end{array}\right) \,, $

where $ \lambda_i$ is an eigenvalue of $ A$.
Since the generalised eigenspaces $ H_\lambda$ are invariant under mapping $ A$, we can reach the block form with respect to a basis of generalised eigenvectors. Using the special cyclic bases

$\displaystyle B^{k_i}v_i,\,\ldots,\,Bv_i,\,v_i,\quad B = A - \lambda_i E \,, $

we obtain blocks of the desired form.

(Authors: Burkhardt/Höllig/Hörner)

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  automatically generated 4/21/2005