mc.0chain.dx | R Documentation |
Extend 0chains of the top left corner to chains for the top left square corner, internal function.
mc.0chain.dx(mo, mo.col, chF0top, F0bot, tol0 = 1e-12)
mo |
mc order |
mo.col |
mc column order, must be less than |
chF0top |
0chains of the top of the mc matrix, a list with one matrix for each chain |
F0bot |
bottom of the top of the mc matrix, a matrix |
tol0 |
tolerance for declaring a vector to be 0 |
For each chain in chF0top
there are two possibilities. Let
v
be the eigenvector in the chain, i.e. F0top
\times
v=0
. If F0bot
\times v
is the zero vector,
then the corresponding chain for the larger matrix is obtained by
extending the vector to size mo x mo
arbitrarily, most
naturally with 0s. Otherwise, if F0bot
\times v
is
not the zero vector, a new eigenvector is (0,...,0,F0bot
\times v
) (there are mo.col zeroes here) and the remaining
members of the chain are obtained from the given chain by
...
. Notice that the eigenvector in chF0top
(after
extension) becomes the second member of the resulting chain, etc.
If the eigenvectors (0,...0,F0bot
\times v_i
) are
linearly independent the job is done. But they may be linearly
dependent. If this is the case some of them need to be dropped and a
transformation is performed to obtain proper Jordan chains.
0chains for the top left mo x mo
corner derived from chF0top
Georgi N. Boshnakov
reduce_chains_simple
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