AIDJEX Bulletin #40 - Polar Science Center - University of Washington

Substituting (36) and (37) into (34) provides one equation for each
characteristic direction. It is
CY,
where 5
is the coordinate associated with X a'
In summary, the characteristic directions and governing equations are
obtained by the following sequence. From the equation (33) determine the
n distinct roots A which define the characteristic directions K These
a
a'
roots arise when the coefficient matrix is singular:
With the roots thus determined, the eigenvectors R are found from (33).
-a
Since the governing equation (32) is homogeneous, the magnitude of R is
-a
immaterial. Finally, along each characteristic direction the solution vector
satisfies an ordinary differential equation (38). In theory, the n equations
may then be integrated, each along its appropriate direction.
EXISTENCE OF CHARACTERISTIC CURVES
For our system of four equations in four unknowns we consider the eigenvalue
problem given by equation (33). Our task is to find four values of X
a
such that the corresponding non-zero R may be found. It is a well-known
-a
property of linear algebraic systems of equations that the coefficient matrix
(XaA-€l)
must be singular. Therefore, we set
det (Aa 4-B) = 0
(39 bis)
and seek roots X
a
substituted into this equation from equations (28) and (29).
that satisfy the equation. The elements of 4 and B may be
Expansion of
the determinant and determination of the roots from the characteristic polynomial
could then follow. However, the special structure of the matrix A4-B
allows a simpler approach. We note that after substitution we may write
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