Observe that all elementary row operations are invertible and any elementary row operation on matrix W [mathematical expression not reproducible] is equivalent to premultiplication
(left multiplication) of W by an appropriate invertible matrix [mathematical expression not reproducible].
sinensis was also reared extensively in premultiplication
of (27) by [[[[phi].sup.mistuning.sub.i]].sup.T] leads to
and a premultiplication
on the left by [Z.sup.*.sub.1] gives
The filtration stage of the "filtered back-projection" method is achieved by premultiplication
of the data measured and is followed by the back-projection operation.
The principal components estimator of the factors consistently estimates [F.sub.t] up to premultiplication
by an arbitrary nonsingular r x r matrix (the analogue of [bar.[LAMBDA]] in the single-factor example); that is, the principal components estimator consistently estimates not the factors, but rather the space spanned by the factors when n and T are large.
Interest in the conjugate gradient method surged again in the 1970s when researchers discovered new variants and successful techniques for preconditioning the problem (i.e., premultiplication
by a carefully chosen easily invertible matrix) to reduce the number of iterations.
Blocked reductions of B to upper triangular form by premultiplication
by, for example, Householder reflections are well known (e.g., see Bischof and Van Loan , Schreiber and Van Loan , and Anderson et al.
Substitution of (10) in (13) and premultiplication
by [Mathematical Expression Omitted] gives the unit revenues per sector (which under perfect competition equals the price of each sector's single input) as the weighted average of that sector's intermediate and final output prices:
This physical force vector is then projected into the reduction space by a premultiplication
with the matrix [[[PHI].bar].sup.T].
by P yields PAv = [mu]Pv which implies v = [mu]Pv (since [mu] [not equal to] 0).
of both sides by [[Beta].sup.-1], the reduced form may be rewritten as: