1. (10 分) 让 a(k 1)
ij 是经过 (k 1) 步高斯消元后 A 的条目。 认为
A(k 1)

2.（15分）设A为严格行对角占优矩阵； IE。

X
j=i

(a) 正如我们在第 13 讲中所做的那样，设 A = D L U，其中 D 是对角线，L 仅包含
A 的对角线下方的条目，U 是对角线上方的条目。 让 G 成为
Gauss-Seidel 方法的迭代矩阵。

(b) 以组件形式写出表达式产生

1. (10 marks) Let a(k 1)
ij be the entries of A after (k 1) steps of Gaussian elimination. Suppose
A(k 1)
is column diagonally dominant; i.e.

If Gaussian elimination without pivoting is used; i.e.

prove that is also column diagonally dominant.
2. (15 marks) Let A be a strictly row diagonally dominant matrix; i.e.
jaiij >
X
j=i
jaij j:
(a) As we did in Lecture 13, let A = D L U, where D is the diagonal, L contains only the
entries below the diagonal of A, and U is the entries above the diagonal. Let G be the
iteration matrix of the Gauss-Seidel method.
Suppose x and y are vectors such that y = Gx: Show that (D L)y = Ux:
(b) Writing out the expression in componentwise form yields

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