Eigenvalues and Eigenvectors

The eigenstructure of is what “unpacks” the matrix exponential and the LTI response.

Definition

• Eigenvalues/eigenvectors: a pair , , such that
• Equivalently:
• Hence is such that is singular, and

Characteristic Polynomial

• singular
• • the “characteristic polynomial” of
• Eigenvalues are the roots of this polynomial:
  • values satisfy this:
• For each eigenvalue , the corresponding eigenvector satisfies
• The char. polynomial also gives the Cayley-Hamilton Theorem

Zero eigenvalues

• If is an eigenvalue of then
• hence is singular and its columns do not define a basis for

Eigenvector Matrix and Diagonalization

• Is there significance also of the eigenvectors of in “unpacking” ? Recall, for each , the eigenvector satisfies
• Let (eigenvector matrix). Then
  • where (diagonal matrix of eigenvalues)

Diagonalization

• (assuming exists!)

• Fact: the eigenvectors for distinct eigenvalues are linearly independent
  • cols of linearly independent
  • exists
• How this is exploited: Dyadic Expansion; when eigenvalues repeat, see Jordan Normal Form

Left and Right Eigenvectors

• Consider
• Row-partition (see Dyadic Expansion):
• Compare with :
  • : “right” eigenvectors (columns of )
  • : “left” eigenvectors (rows of )
• Equivalently:
  • are the ordinary (right) eigenvectors of

NOTE

• and have the same char. poly, hence the same eigenvalues :
  • , so

Bi-Orthonormality

• Now , or
• so that

Bi-orthonormality

• Hence
• Thus and are “bi-orthonormal” sets