In the eigenvalue problem formulated in equation (2), q is an n x 1 unknown vector known as the eigenvector, or characteristic vector, or latent vector; [lambda] is an unknown scalar known as the eigenvalue, or characteristic root
, or latent root (Hoy, Livernois, McKenna, Rees, and Stengos, 1996).
The proportion of variation attributed to a particular principal component is obtained by dividing the associated characteristic root
by the sum of all the characteristic roots
Heck, Charts of Some Upper Percentage Points of the Distribution of the Largest Characteristic Root
When the characteristic root
for a regime is zero, there is an immediate jump to the attractor for that regime.
Letting [Lambda]be the characteristic root
of the dynamic system and linearing the system around the stationary equilibrium, we then have the following characteristic equation:
It is easy to verify from the entries in Table II that only one characteristic root
, derived from the correlation matrix of the five diffusion indexes, is greater than 0.
A permanent increase in productivity level ([Tau]) and a drop in interest rates (r) increase the new steady state value of capital and change the value of the characteristic root
, which, when the changes occur, lead to the upward jumps in the shadow price of capital (q), investment (i), output (f(k)), and consumption (c) instantaneously (and permanently).
This procedure relies on relationships between the rank of a matrix and its characteristic roots
So now the characteristic roots
are given by the following two equations
The Johansen 1988 method relies on the relationship between the rank of a matrix and its characteristic roots
2] are constants, which we call the characteristic roots
and a is the minimum among the real parts of characteristic roots
to the right of -1.