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Then A
B
.
B .
Also, notice that the Kronecker product can be calculated for any
two matrices. Recall that in order to calculate the usual product of two
matrices, the number of columns of the first matrix has to be equal to
the number of rows of the second matrix.
Exercise : Calculate B
A . Show that it is not equal to A
Modeling the variance-covariance matrix V for the matrix
normal distribution
Let us consider a two-dimensional object with K landmarks. We know
that the variance-covariance matrix corresponding to the measure-
ment error model described above is of dimension 2 K
2 K . In some
special situations, the matrix V can be written in the Kronecker prod-
uct form. We describe some of these situations below. In the next
chapter, we argue that these models are mathematically convenient
and biologically reasonable for modeling biological variability.
Situation 1: Suppose the measurement errors at all land-
marks and along all directions are identical to each other. In
addition, suppose that they are also uncorrelated to each
other. In this situation the variance-covariance matrix V has
the same quantity, say
, along the diagonal and all off-diag-
onal entries are zero. Such a matrix may be written as
2
I 2K .
It can also be written equivalently in the Kronecker product
form as
2
I 2 . This model is sometimes known as the
“isotropic errors” model.
2
I K
Situation 2: The assumption of isotropic errors may seem
unrealistic in practice. Suppose the measurement errors at
different landmarks are correlated, but that the error along
different axes are uncorrelated and equal. The covariance
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