## Eigenvector

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 Self-Adjoint 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,
 (12)

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.

Arfken, G. Eigenvectors, Eigenvalues.'' §4.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 229-237, 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. 449-489, 1992.