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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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