課程名稱︰分析導論優二
課程性質︰數學系選修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰103/3/24
考試時限（分鐘）：50
試題 :
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.