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releigen(h,e [,maxit:N, nonconvok:T]), h and e symmetric REAL matrices
  with no MISSING values, e positive definite, integer N > 0

Keywords: matrix algebra
releigen(H,E) computes an eigenvector/eigenvalue decomposition of the
symmetric matrix H relative to the symmetric positive definite matrix E.
Arguments H and E must both be p by p symmetric matrices with no MISSING

The value returned is structure(values:Vals,vectors:Vecs), where Vals is
the length p vector of relative eigenvalues in decreasing order and Vecs
is a p by p matrix whose columns are the relative eigenvectors.

If H and E are MANOVA hypothesis and error matrices, respectively, you
can use the relative eigenvalues to compute several standard
multivariate hypothesis tests, and the elements of the relative
eigenvectors are the coefficients of the MANOVA canonical variables
associated with the hypothesis.

  Cmd> releigs <- releigen(H,E) # H and E p by p

  Cmd> v <- releigs$values; u <- releigs$vectors

u is a p by p matrix and v is a vector of length p with elements v[1] >=
v[2] >= ... >= v[p], such that
     H %*% u        = E %*% u %*% dmat(v)
     u' %*% H %*% u = dmat(v)
     u' %*% E %*% u = Ip = p by p identity matrix

On the vector level, this means
     H %*% u[,j] = v[j] * E %*% u[,j], j = 1,...,p
     u[,j]' %*% H %*% u[,j] = v[j], j = 1,..,p
     u[,j]' %*% H %*% u[,k] = 0, j != k
     u[,j]' %*% E %*% u[,j] = 1, j = 1,..,p
     u[,j]' %*% E %*% u[,k] = 0, j != k

dmat(v) is the diagonal matrix with elements of v down the diagonal.

See eigen() for information on keyword phrases 'maxit:N' and

See also det(), eigen(), eigenvals(), trideigen() and releigenvals().

Gary Oehlert 2003-01-15