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Singular Value Decomposition
Singular Value Decomposition
4 questions
Difficulty 2-5
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Foundation
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2 foundation
2 intermediate
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foundation (2/10)
spot the error
A student writes: 'The SVD of a matrix
A
∈
R
m
×
n
is
A
=
U
Σ
V
where
U
is
m
×
m
orthogonal,
Σ
is
m
×
n
diagonal, and
V
is
n
×
n
orthogonal.' What is the error?
Hide and think first
A.
The dimensions of
Σ
are wrong: it should be
min
(
m
,
n
)
×
min
(
m
,
n
)
square diagonal, not
m
×
n
rectangular with zero padding
B.
U
and
V
should be described as unitary rather than orthogonal, because the SVD requires the stronger condition of complex unitarity for correctness
C.
Missing the transpose on
V
: should be
A
=
U
Σ
V
T
. The columns of
V
are right singular vectors, so
V
T
is needed for the product to reconstruct
A
D.
The singular values in
Σ
should be complex numbers in general, not necessarily real, since
A
could have complex eigenvalues even when real-valued
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