Page 149 - Textos de Matemática Vol. 39

QUASIASSOCIATIVE ALGEBRAS 139
This expression is equal to
n φ(q, q−1, qm−1)
φ(q−1,q,q−1)xqm φ(q−1,q,m−1)Eqm+ q,m=1,q̸=i,j
n φ(i, j−1, jm−1)
φ(j−1,j,j−1)xjm φ(j−1,j,m−1)Eim+ m=1
n φ(j, i−1, im−1)
φ(i−1,i,i−1)xim φ(i−1,i,m−1)Ejm. m=1
As φ(l,l−1,l)φ(l−1,l,m−1)=φ(l,l−1,lm−1), for any l we conclude that
n q,m=1,q̸=i,j
n φ(i, j−1, jm−1) xqmEqm + xjm φ(j,j−1,jm−1)Eim+
PijφX =
xim φ(i,i−1,im−1)Ejm. (2) Eijφ(α)X = (Iφ + αEij)X = X + (αEij)X =
n φ(i, j−1, jm−1)
X+ αxjm φ(j−1,j,m−1)Eim.
m=1
n m=1
m=1
φ(j, i−1, im−1)
It easily seen that, for any α ∈ K and X ∈ Mn,φ(K), we have Eijφ(α)Eijφ(−α) = Eijφ(−α)Eijφ(α) = Iφ and Eijφ(−α)(Eijφ(α)X) = X.
Consider X = [xij] ∈ Mn,φ(K) and suppose that xim and xjm are two nonzero entries in the mth column of X. From the previous lemma we know
that the matrix E (−xi,m φ(j−1,j,m−1) )X has zero (im) entry. ijφ xj,m φ(i,j−1 ,jm−1 )
This combined with the fact that multiplying Pijφ by X ‘interchanges’ the rows of X tell us how using deformed elementary matrices we can obtain in Mn,φ(K) a ‘quasi − LU’ decomposition of the matrix X.
In the following lemmas of this section X denotes an arbitrary matrix in Mn(K, φ), α1, α2, ..., αr are elements of K and we will consider products of matrices
Ei1j1φ(α1),Ei2j2φ(α2),...,Eirjrφ(αr), whereip >jp, p=1,···,r and j1 ≤j2 ≤···≤jr.