A right eigenvector satisfies

(1) 
where is a column Vector. The right Eigenvalues therefore satisfy

(2) 
A left eigenvector satisfies

(3) 
where is a row Vector, so

(4) 

(5) 
where
is the transpose of .
The left Eigenvalues satisfy

(6) 
(since
) where
is the Determinant
of A. But this is the same equation satisfied by the
right Eigenvalues, so the left and right Eigenvalues are the same. Let be a Matrix formed by the columns of the right eigenvectors and be a Matrix formed by the
rows of the left eigenvectors. Let

(7) 
Then

(8) 

(9) 
so

(10) 
But this equation is of the form
where
is a Diagonal Matrix, so it must be true
that
is also diagonal. In particular, if A is a Symmetric Matrix, then the
left and right eigenvectors are transposes of each other. If A is a SelfAdjoint Matrix, then the left and
right eigenvectors are conjugate Hermitian Matrices.
Given a Matrix A with eigenvectors , , and and corresponding
Eigenvalues , , and , then an arbitrary Vector
can be written

(11) 
Applying the Matrix A,
so

(13) 
If
, it therefore follows that

(14) 
so repeated application of the matrix to an arbitrary vector results in a vector proportional to the Eigenvector
having the largest Eigenvalue.
See also Eigenfunction, Eigenvalue
References
Arfken, G. ``Eigenvectors, Eigenvalues.'' §4.7 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 229237, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Eigensystems.'' Ch. 11 in
Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 449489, 1992.
© 19969 Eric W. Weisstein
19990525