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Arnold's Trivium
1991 Arnold's Trivium
37
Problem 37
Problem 37
Source:
July 1, 2010
Problem Statement
Prove that in Euclidean space the surfaces
(
(
A
−
λ
E
)
−
1
x
,
x
)
=
1
((A-\lambda E)^{-1}x,x)=1
((
A
−
λ
E
)
−
1
x
,
x
)
=
1
passing through the point
x
x
x
and corresponding to different values of
λ
\lambda
λ
are pairwise orthogonal (
A
A
A
is a symmetric operator without multiple eigenvalues).
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