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
where H = [([h.sup.ij]).sub.NxN] is a lower
triangular matrix specified as follows: