课程名称︰分析导论优二
课程性质︰数学系选修
课程教师︰王振男
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰103/3/24
考试时限（分钟）：50
试题 :
1. (10%) Prove that the implicit function theorem implies the inverse function
theroem.
2. (10%) (a) Let A be a n × n real symmetric matrix. Let
T 2 n
λ = max{x Ax : ||x|| = x‧x = 1, x ∈ IR }.
1
Show that λ is the largest eigenvalue of A.
1
(b) Assume that λ is simple with eigenvalue v. Prove that
1
T 2
λ = max{x Ax : ||x|| = 1, x‧v = 0}
2
is also an eigenvalue of A.