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Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
5 questions
Difficulty 2-4
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counterexample
A student claims: 'If
A
is a real symmetric matrix, then all its eigenvalues are positive.' Which counterexample disproves this?
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A.
A
=
(
0
−
1
1
0
)
: this has purely imaginary eigenvalues
±
i
, disproving the claim about positive eigenvalues
B.
The zero matrix
A
=
0
: it is symmetric with all eigenvalues equal to zero, which are not positive, disproving the general claim
C.
A
=
(
1
3
2
4
)
: this has one negative eigenvalue, disproving the claim that symmetric matrices have positive eigenvalues
D.
A
=
diag
(
1
,
−
1
)
: symmetric with eigenvalues
1
and
−
1
. Symmetry guarantees real eigenvalues, not positive ones; that requires positive definiteness
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