2.3 Left Eigenvectors and Jordan Chains

Let us consider the eigenvalue problem for the transposed matrix

(2.27)

Since

(2.28)

the characteristic equations for the matrices A and A ^T coincide. The matrix A ^T has the same Jordan canonical form as the matrix A, see [Gantmacher (1998)]. Therefore, eigenvalues of the matrices A and A ^T are equal together with their algebraic, geometric, and partial multiplicities.

The eigenvalue problem for the matrix A ^T after the transposition takes the form

(2.29)

where v is called the left eigenvector corresponding to the eigenvalue ?, in contrast to the eigenvector u called the right eigenvector.

If an eigenvalue ? is simple, then the left eigenvector is determined up to a nonzero scaling factor. Assuming that the right eigenvector u is given, we can define the left eigenvector of the simple eigenvalue uniquely by means of the normalization condition

(2.30)

Notice that if u ? is a right eigenvector for an eigenvalue ? ?, and v is a left eigenvector for an eigenvalue ? ? ? ?, then [Gantmacher (1998)]

(2.31)

Equations of the Jordan chain (2.4) for a nonderogatory eigenvalue ? of the matrix A ^T after transposition take the form

(2.32)

The vectors v ₀, ..., v _{k ?1} are called the left Jordan chain for the eigenvalue ? as opposed to the right Jordan chain u ₀