a_{11} 
a_{12} 
a_{13} 
a_{14} 

matrix A = 
a_{21} 
a_{22} 
a_{23} 
a_{24} 
a_{31} 
a_{32} 
a_{33} 
a_{34} 
a_{13}  
a_{·}_{3} =  a_{23} 
a_{13} 
where ·
stands for all the elements in the corresponding dimension (this usage should
not be confused with multiplication, where the · will be between two arguments,
e.g. a·b, meaning a × b).
The transpose of A
=
a_{11}  a_{21}  a_{31}  
A´=  a_{12}  a_{22}  a_{32} 
a_{13}  a_{23}  a_{33}  
a_{14}  a_{24}  a_{34} 
Note
that the first row of A becomes the first column of A´,
while the first column of A becomes the first row of A´.
A
is a square matrix if m = n. Elements of A, a_{hi},
such that h = i are the diagonal elements of A. The
sum of the diagonal elements of a square matrix is its trace.
Square matrices can
be of particular kinds, e.g.
A
is a symmetric square matrix if A´ = A.
A is a
diagonal matrix if the offdiagonal
elements are all zero,
a_{11}  0  0  
e.g.  0  a_{22}  0 
0  0  a_{33} 
A vector is an ordered set of real numbers (scalars) that can be written as
or as a´ = (a_{1},a_{2},a_{3}, ...a_{n}). Geometrically, the elements of a can be thought of as coordinates of a point in ndimensional space. We will restrict our discussion of vectors to what are called position vectors, in that they represent the direction and distance from the origin of an ndimensional coordinate system to the point in space defined by the elements of a (see illustration at right). The 12 2element (x_{1}, x_{2}) position vectors shown below can be thought of as making up a 12 × 2 matrix A. 

Addition (and subtraction)

Matrix
multiplication is based on the scalar products of vectors:
a_{11}  b_{11}  b_{11}  
for a =  a_{21}  and b =  b_{21}  a´ b = a_{11} a_{21} a_{31} ·  b_{21}  = a_{11}·b_{11} + a_{21}·b_{21} + a_{31}·b_{31} 
a_{31}  b_{31}  b_{31} 
This is a scalar number,
equivalent to a matrix of order 1 × 1, that is, with rows equal to those of
the first matrix and columns equal to those of the second matrix.
Why is the square
root of a´a equal to the length of vector a?
Consider the vector a´ shown at the right; its length (denoted a´) is given by generalizing the Pythagorean theorem to three dimensions (or more, depending on the number of elements in a).
a´a =
a_{11}·a_{11} + a_{21}·a_{21} + a_{31}·a_{31}

Multiplication
of entire matrixes is thus defined: for
matrices A and B, their product AB
can be thought of as containing elements that are the scalar products of the
row vectors of A and the column vectors of B.
In order for this to be possible, the number of columns in A
must equal the number of rows in B. When this is the case,
A and B are said to conform. Matrix
multiplication is possible if and only if the number of columns of the
first matrix equals the number of rows of the second.
Based on this definition,
the results of matrix multiplication depend on the order in which matrices
are multiplied.
For matrices A
and B,
a_{11}  a_{12}  a_{13}  (a_{11}·b_{11}+a_{12}·b_{21}+a_{13}·b_{31})  (a_{11}·b_{12}+a_{12}·b_{22}+a_{13}·b_{32})  
a_{21}  a_{22}  a_{23}  b_{11}  b_{12}  (a_{21}·b_{11}+a_{22}·b_{21}+a_{23}·b_{31})  (a_{21}·b_{12}+a_{22}·b_{22}+a_{23}·b_{32})  
AB =  a_{31}  a_{32}  a_{33}  ·  b_{21}  b_{22}  =  (a_{31}·b_{11}+a_{32}·b_{21}+a_{33}·b_{31})  (a_{31}·b_{12}+a_{32}·b_{22}+a_{33}·b_{32}) 
a_{41}  a_{42}  a_{43}  b_{31}  b_{32}  (a_{41}·b_{11}+a_{42}·b_{21}+a_{43}·b_{31})  (a_{41}·b_{12}+a_{42}·b_{22}+a_{43}·b_{32})  
a_{51}  a_{52}  a_{53}  (a_{51}·b_{11}+a_{52}·b_{21}+a_{53}·b_{31})  (a_{51}·b_{12}+a_{52}·b_{22}+a_{53}·b_{32}) 
or, more compactly,
(a_{1·} · b._{1})  (a_{1·} · b._{2}) 
(a_{2·} · b._{1})  (a_{2·} · b._{2}) 
(a_{3·} · b._{1})  (a_{3·} · b._{2}) 
(a_{4·} · b._{1})  (a_{4·} · b._{2}) 
(a_{5·} · b._{1})  (a_{5·} · b._{2}) 
or (for N=3)
(a_{1i}·b_{i1})  (a_{1i}·b_{i2}) 
(a_{2i}·b_{i1})  (a_{2i}·b_{i2}) 
(a_{3i}·b_{i1})  (a_{3i}·b_{i2}) 
(a_{4i}·b_{i1})  (a_{4i}·b_{i2}) 
(a_{5i}·b_{i1})  (a_{5i}·b_{i2}) 
where
signifies summation over i=1...N.
What would be the
order of A´ A?  of AA´ ?
Some useful geometric
properties of matrix multiplication are illustrated HERE
(esp. pp. 13).
The basic data matrix that is used in (numerical) phenetics is one in which one dimension (rows, say) represents the operational taxonomic units (OTUs) while the other dimension (columns) represents the characters for which data have been collected. For example, consider the data in Table 1 (below) as representing a description of 11 races of bean rust, based on their reactions when inoculated onto a suite of differential bean cultivars.
Table 1. Disease reactions for 11 Jamaican rust (Uromyces appendiculatus var. appendiculatus) isolates (J1J20) inoculated on 10 bean (Phaseolus vulgaris) cultivars (first six, US differential vars.; last four, Jamaican landraces): 1, susceptible (sporulation); 0, immune (no sporulation). Data from M. Shaik (1983).  
GGW  814  765  111  643  181  PR  RR  JR  MK  
J1  1  0  0  1  0  1  0  0  1  0 
J3  1  0  0  0  0  1  0  0  1  0 
J20  1  1  0  1  0  1  0  1  1  0 
J11  1  1  1  1  0  1  1  1  1  1 
J12  1  1  1  1  0  1  1  1  1  1 
J13  1  1  1  1  0  1  1  1  1  1 
J5  1  1  1  1  1  1  1  1  1  1 
J6  1  1  1  1  1  1  1  1  1  1 
J10  1  1  1  1  1  1  1  1  1  1 
J14  0  1  1  1  1  1  1  1  1  1 
J15  0  1  1  1  1  1  1  1  1  1 
Covariance and correlation (page 4 ff.)
Euclidean distance (first 2 pages only)
Measures for binary and multistate data
Measures for mixed data
Legendre, P. & L. Legendre (1998). Numerical Ecology, 2nd ed. New York, SpringerVerlag.
Manly, B. F. J. (1994). Multivariate statistical methods  A primer, 2nd ed. London, Chapman & Hall.
Podani, J. (2000). Introduction to the exploration of Multivariate biological data. Leiden, Backhuys.
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