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matrix corresponding to such a measurement error structure
can be written as
K
is the covariance matrix
between landmarks. Remember that in order for
I
2
where
K
K
to be a
covariance matrix, it must be a
K
K
square symmetric
matrix.
Situation 3:
In the above model, we may want to include dif-
ferent magnitudes of variability along the two axes and also
correlations between the axes. In this case, the covariance
matrix may be written as
K
2
where
K
is a
K
K
square,
symmetric matrix, and
2
is a 2 by 2 square, symmetric
matrix.
Remarks:
1.
These examples of covariance matrices for measurement
error are not exhaustive. There are certain covariance struc-
tures that are not captured by the above matrices.
2.
Generalization to three-dimensional object entails replacing
all 2 x 2 matrices in the above discussion by 3 x 3 matrices.
Strictly speaking, the matrices
K
and
D
should also be pos-
itive semidefinite, that is, all the eigenvalues of these
matrices should be non-negative. See Barnett (1990) for
details on positive semidefinite matrices.
3.
2.8.3 Effect of rotation and translation on the matrix normal distribution
The following observations are standard results from the theory of
matrix valued Normal distribution. See Arnold (1981, page 312) for
mathematical details. These results are used extensively in
Chapters
3
and
4
. For this reason, we present them here. Readers who are not
comfortable with the mathematical aspects should at least read
through this part and familiarize themselves with the notation and the
intuitive meaning of these results. This will prepare the reader for the
Let us first try to understand
intuitively
what we mean by the
statement an observation X, “comes from” or equivalently, “follows” a
normal distribution. Let us say that X corresponds to “height of a per-
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