EIGEN Call

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EIGEN Call

computes eigenvalues and eigenvectors

CALL EIGEN( eigenvalues, eigenvectors, A) <VECL="vl">;

where A is an arbitrary square numeric matrix for which eigenvalues and eigenvectors are to be calculated.

The EIGEN call returns the following values:

eigenvalues: a matrix to contain the eigenvalues of the input matrix.
eigenvectors: names a matrix to contain the right eigenvectors of the input matrix.
vl: is an optional n ×n matrix containing the left eigenvectors of A in the same manner that eigenvectors contains the right eigenvectors.

The EIGEN subroutine computes eigenvalues, a matrix containing the eigenvalues of A arranged in descending order. If A is symmetric, eigenvalues in the n ×1 vector containing the n real eigenvalues of A. If A is not symmetric (as determined by the criterion described below) eigenvalues is an n ×2 matrix containing the eigenvalues of the n ×n matrix A. The first column of A contains the real parts, Re( $\lambda)$ , and the second column contains the imaginary parts Im $(\lambda)$ .Each row represents one eigenvalue, ${Re}(\lambda) + i{Im}(\lambda)$ .Complex conjugate eigenvalues, Re $(\lambda)+- i{Im}(\lambda)$ ,are stored in standard order; that is, the eigenvalue of the pair with a positive imaginary part is followed by the eigenvalue of the pair with the negative imaginary part.

The EIGEN subroutine also computes eigenvectors, a matrix. If A is symmetric, then eigenvectors has orthonormal column eigenvectors of A arranged so that the matrices correspond; that is, the first column of eigenvectors is the eigenvector corresponding to the largest eigenvalue, and so forth. If A is not symmetric, then eigenvectors is an n ×n matrix containing the right eigenvectors of A. If the eigenvalue in row i of eigenvalues is real, then column i of eigenvectors contains the corresponding real eigenvector. If rows i and i+1 of eigenvalues contain complex conjugate eigenvalues Re $(\lambda)+- i{Im}(\lambda)$ ,then columns i and i+1 of eigenvectors contain the real, v, and imaginary, u, parts, respectively, of the two corresponding eigenvectors .

The eigenvalues of a matrix A are the roots of the characteristic polynomial, which is defined as p(z) = det(zI- A). The spectrum, denoted by $\lambda(A)$ , is the set of eigenvalues of the matrix A. If $\lambda(A)=\{\lambda_1, ... ,\lambda_n\}$ , then $\det(A) = \lambda_1 \lambda_2 ... \lambda_n$ .

The trace of A is defined by

${\rm tr}(A) = \sum_{i=1}^n a_{ii}$

and tr $(A)= \lambda_1 + ... + \lambda_n$ .

An eigenvector is a nonzero vector, x, that satisfies $A{x}= \lambda{x}$ for $\lambda \in \lambda(A)$ . Right eigenvectors satisfy $A{x}= \lambda{x}$ , and left eigenvectors satisfy $x^'A= \lambda{x}^'$ .

The following are properties of the unsymmetric real eigenvalue problem, in which the real matrix A is square but not necessarily symmetric:

The eigenvalues of an unsymmetric matrix A can be complex. If A has a complex eigenvalue ${Re}(\lambda) + i{Im}(\lambda)$ , then the conjugate complex value ${Re}(\lambda)-i{Im}(\lambda)$ is also an eigenvalue of A.
The right and left eigenvectors corresponding to a real eigenvalue of A are real. The right and left eigenvectors corresponding to conjugate complex eigenvalues of A are also conjugate complex.
The left eigenvectors of A are the same as the complex conjugate right eigenvectors of A'.

The three routines, EIGEN, EIGVAL, and EIGVEC, use the following test of symmetry for a square argument matrix A:

Select the entry of A with the largest magnitude:
$a_{max} = \max_{i,j=1, ... ,n} | a_{i,j}|$
Multiply the value of a_max with the square root of the machine precision $\epsilon$ . (The value of $\epsilon$ is the largest value stored in double precision that, when added to 1 in double precision, still results in 1.)
The matrix A is considered unsymmetric if there exists at least one pair of symmetric entries that differs in more than $a_{max}\sqrt{\epsilon}$ ,
$| a_{i,j}-a_{j,i}| \gt a_{max} \sqrt{\epsilon}$

If A is symmetric, the result of the statement

   call eigen(m,e,a);

has the properties

$A*E & = & E*{diag}(M) \E^'*E & = & I(N)$

that is,

E' = inv(E)

so that

A = E*diag(M)*E' .

The QL method is used to compute the eigenvalues (Wilkinson and Reinsch 1971).

In statistical applications, nonsymmetric matrices for which eigenvalues are desired are usually of the form E^-1 H, where E and H are symmetric. The eigenvalues L and eigenvectors V of E^-1H can be obtained by using the GENEIG subroutine or as follows:

   f=root(einv);
   a=f*h*f';
   call eigen(l,w,a);
   v=f'*w;

The computation can be checked by forming the residuals:

   r=einv*h*v-v*diag(l);

The values in R should be of the order of round-off error.

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