Since the matrix A is symmetric positive definite, we can decompose it into a diagonal matrix [D.sub.A], a strictly lower triangular matrix
[L.sub.A] , and a strictly upper triangular matrix
where S [member of] [R.sup.nxn] is a positive definite symmetric matrix and T [member of] [R.sup.nxn] is a unity upper triangular matrix
where [e.sub.1] is the first s column of the (m + 1)s x (m + 1)s unit matrix (the size changes with m), [[rho].sub.0] is an upper triangular matrix
obtained in Arnoldi's initialization step, and [t.sub.m] is the "block coordinates" of [X.sub.m] - [X.sub.0] with respect to the block Arnoldi basis.
Based on the triangular matrix
A of size m x m, where [A.sub.i,j] = 0 for i > j, it is indicated that the set of m x m triangular matrices exists in [R.sub.max], but the operator [cross product] is not commutative.
Adams  defined that the four-dimensional infinite matrix A = ([a.sub.mnkl]) is called a triangular matrix
if [a.sub.mnkl] = 0 for k > m or l > n or both.
As applications, (m, n)--Jordan centralizers on (block) upper triangular matrix
algebras and nest algebras are centralizers.
A is called triangular NSM if it is either neutrosophic soft upper triangular or neutrosophic soft lower triangular matrix
(4) is a lower block triangular matrix
, the most efficient and robust solver is a direct block triangle matrix solver.
where [H.sub.1,0] is the upper triangular matrix
appearing in the QR decomposition of Y, i.e., [[V.sub.1], [H.sub.1,0]] = QR(Y).
If A [member of] B(X) and M is an invariant (assumed, as before, to be closed) subspace of A, then A has an upper triangular matrix
As S and [??] are the lower triangular matrix
, it is known that [??] is the matrix composed of elements of the former l rows and former l columns of S.
where H = [([h.sup.ij]).sub.NxN] is a lower triangular matrix
specified as follows: