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 [L.sup.H.sub.A].

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 [7] 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 representation

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: