Given two vectors f
and g the following notations are used:
| f>>g | if fi>gi for all i. |
| f>g | if fi >=gi for all i, and fi>gi for some i. |
| f>=g | if fi >=gi for all i. |
| f=g | if fi=gi for all i. |
| f<>g | if fi <>gi for some i. |
Given two matrices
A and B the following notations are used:
| A>>B | if Aij>Bij for all i and j. |
| A>B | if Aij>=Bij for all i and j, and Aij>Bij for some i and j. |
| A>=B | if Aij>=Bij for all i and j. |
| A=B | if Aij=Bij for all i and j. |
| A<>B | if Aij<>Bij for some i and j. |
In particular
| If f>>0 | then we call f positive. |
| If f>0 | then we call f semi positive. |
| If f>=0 | then we call f nonnegative. |
| If f=0 | then we call f zero. |
| If f<>0 | then we call f nonzero. |
The same terminology is used for matrices
Every reducible matrix
N can be rewritten by simultaneous row and column permutations in the form:
 |
Where P is a permutation
matrix as obtained by permuting the rows or columns of the identity matrix.
The matrices on the main diagonal are all square and not reducible. For
the off-diagonal sub matrices we have at least one nonzero sub matrix in
every row. (I) is known as the 'Normal form'. A matrix is called irreducible
if it is not reducible. A matrix with every block, except those on the
main diagonal, zero is called 'fully reducible' or 'block diagonal'. In
what follows we shall, at certain stages, use a special case of the 'Normal
form I', therefore the 'Normal form II' is defined as
 |
Throughout this paper we only consider nonnegative matrices (A>=0).
When we have Ax=λx, with the vector x<>0 and λ a scalar (real or complex), then we call λ an eigenvalue of A. The vector x is called the eigenvector associated with λ.
The dominant eigenvalue λ* is that eigenvalue where |λ|<=λ* for any other eigenvalue λi of A. Note that λ* is always real and nonnegative. A Perron eigenvector is an eigenvector which belongs to the eigenvalue λ*. The multiplicity of λ* as a zero of the characteristic polynomial of A is called the algebraic multiplicity of the eigenvalue λ*. The dimension of eigenspace that is spanned by the corresponding eigenvectors is called the geometric multiplicity. The geometric multiplicity is always smaller than or equal to the algebraic multiplicity.
All the blocks Aii are irreducible. This implies that each of them has an unique (up to scalar multiple) real positive dominant eigenvector. We shall denote the right real positive dominant eigenvector of block Aii by qi, and the left real positive dominant eigenvector by yi. The left and right eigenvector of a reducible matrix A will be denoted by respectively v and w. v and w are of course partitioned according to the partition of A. The dominant eigenvalue of A is denoted by λ*. The dominant eigenvalue of Aii is denoted by λi.