# premultiplication

## premultiplication

[‚prē·məl·tə·plə′kā·shən]
(mathematics)
In multiplying a matrix or operatorBby another matrix or operatorA, the operation that results in the matrix or operatorAB. Also known as multiplication on the left.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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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].
Premultiplication of (27) by [[[[phi].sup.mistuning.sub.i]].sup.T] leads to
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].
Premultiplication by P yields PAv = [mu]Pv which implies v = [mu]Pv (since [mu] [not equal to] 0).
After premultiplication of both sides by [[Beta].sup.-1], the reduced form may be rewritten as:

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