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

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 values. 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. After 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 'nonconvok:T'. See also det(), eigen(), eigenvals(), trideigen() and releigenvals().

Gary Oehlert 2003-01-15