Biology Reference

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01 1

10

FM
()

.

1

1

1 41

.

0

where columns represent landmarks 1, 2, and 3 read from left to right,

and rows represent landmarks 1, 2, and 3 read from top to bottom. The

matrix is square because the number of columns is equal to the number

of rows. Each cell represents the distance between the landmark that

heads a row and the landmark that heads a column. The cell in the

upper left-hand corner represents the linear distance between land-

mark 1 and itself. The second cell in the first row represents the

distance between landmarks 1 and 2, while the third cell in the first row

represents the distance between landmarks 1 and 3. Notice that the

entries on the diagonal are all 0 because the distance between a land-

mark and itself is zero. The matrix is symmetric, meaning that the

entry for the (
i, j
)-th cell is equal to the entry for the (
j, i
) - t h cell. The

first cell in the second row represents the distance between landmarks

2 and 1, which is equal to the value entered in the second cell in the first

row, the distance between landmarks 1 and 2.

Since this matrix is symmetric, we need not write it in full matrix

format but can abbreviate it by collecting only the above-diagonal ele-

ments and writing those numbers as a vector.

When written as a vector, the elements to the right of the first diago-

nal element in row 1 are written in order first, followed by the

elements to the right of the diagonal in row 2 and so on. Unless we

specify otherwise, the form matrix is always written as a vector.

3.7 The ability to estimate the mean form and variance

In this section, we provide a verbal description of two key results

regarding the estimation of the mean form and variance. These results

are proven in
Part 2
of this chapter. The discussion in this section is

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