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In-Depth Information
various concepts. In general, we can measure the similarity of users by their cosine distance
in concept space.
EXAMPLE
11.11 For the case introduced above, note that the concept vectors for Quincy
and Joe, which are [2.32, 0] and [1.74, 0], respectively, are not the same, but they have ex-
actly the same direction. That is, their cosine distance is 0. On the other hand, the vectors
for Quincy and Jill, which are [2.32, 0] and [0, 5.68], respectively, have a dot product of
0, and therefore their angle is 90 degrees. That is, their cosine distance is 1, the maximum
possible.
□
11.3.6
Computing the SVD of a Matrix
The SVD of a matrix
M
is strongly connected to the eigenvalues of the symmetric matrices
M
T
M
and
MM
T
. This relationship allows us to obtain the SVD of
M
from the eigenpairs of
the latter two matrices. To begin the explanation, start with
M
=
U
Σ
V
T
, the expression for
the SVD of
M
. Then
M
T
= (
U
Σ
V
T
)
T
= (
V
T
)
T
Σ
T
U
T
=
V
Σ
T
U
T
Since Σ is a diagonal matrix, transposing it has no effect. Thus,
M
T
=
V
Σ
U
T
.
Now,
M
T
M
=
V
Σ
U
T
U
Σ
V
T
. Remember that
U
is an orthonormal matrix, so
U
T
U
is the
identity matrix of the appropriate size. That is,
M
T
M
=
V
Σ
2
V
T
Multiply both sides of this equation on the left by
V
to get
M
T
MV
=
V
Σ
2
V
T
V
Since
V
is also an orthonormal matrix, we know that
V
T
V
is the identity. Thus
M
T
MV
=
V
Σ
2
(11.6)
Since Σ is a diagonal matrix, Σ
2
is also a diagonal matrix whose entry in the
i
th row and
familiar. It says that
V
is the matrix of eigenvectors of
M
T
M
and Σ
2
is the diagonal matrix
whose entries are the corresponding eigenvalues.
