Jordan Normal Form

The “closest to diagonal” LTI coordinate transformation when has repeated eigenvalues.

Motivation

• Recall: when all are distinct we can use to produce a purely diagonal (modal form)
• Another reason to find an alternate for LTI: repeated (real) eigenvalues
  • (repeated poles also give terms in : see Poles and Eigenvalues)

Jordan Normal Form

• Suppose
  • i.e. repeated times; distinct
• Then such that has the form

Jordan Normal Form

• • the first rows/columns hold the Jordan blocks of ; zeros elsewhere
• i.e.
• the “Jordan normal form”

• where each is an “Jordan block”:
• Constraint:

Multiplicities

• Terminology:
  • : “algebraic multiplicity” of
  • : “geometric multiplicity”
• Fact:
  • = # of linearly independent eigenvectors associated with
  • (by rank-nullity)

Special Cases

1. :
   • linearly independent eigenvectors for
   • Let be these eigenvectors, and
   • Then is purely diagonal:
   • here and
2. :
   • only 1 Jordan block, which must be
   • where (see Jordan block exponential)
3. For : we know how many Jordan blocks, but not necessarily their sizes

Generalized Eigenvectors

• Note for :
• The remaining columns come from a “chain of generalized eigenvectors”, satisfying
• These “chains” generate, in a complicated fashion, the exact sizes of each block
• We do at least have the constraint
  • which can sometimes uniquely determine the structure of , i.e.
  • But if , we could have
    • or
• Which case we’re in depends on the exact details of , and is very sensitive to small changes in the entries of

Sensitivity

Sensitivity of the Jordan Form

• If :
  • repeated eigenvalue, ,
• If :
  • (non-repeated eigenvalues)
• The calculation is very sensitive to even the smallest perturbation in

NOTE

• Matlab’s jordan function is useful, but doesn’t always work