mc_0chains: Generate eigenchains for zero eigenvalues

In mcompanion: Objects and Methods for Multi-Companion Matrices

Generate eigenchains for zero eigenvalues

Description

Generate eigenchains for zero eigenvalues of multi-companion matrices

Usage

mc_0chains(dim, mo, mo.col, vec0, flagtriang = TRUE)

Arguments

dim	dimension of the matrix
mo	multi-companion order
mo.col	number of (non-zero) structural columns
vec0	a list of Jordan chains for non-structural zero eigenvalues, see details

flagtriang

triangulase the vectors?

Details

Non-structural eigenvectors for zero eigenvalues may be specified with the vec0 argument. It is a list with one element for each non-structural 0 eigenvalue. This element is a matrix with the seeding chain for the corresponding chain (!!! edit this !!!) with enough number of rows (at least the multi-companion order). This is often a zero chain for a mc matrix of smaller or larger size. In that case the vectors in the chain are extended or shrinked, respectively. However, extending and shrinking of 0-chains is different from the ordinary case of non-zero eigenvalues and involves increasing or decreasing the number of vectors in the chain.

If vec0 is missing or equal to list() the function assumes that the only 0 eigenvalues are the structural ones, i.e. that there are no 0 eigenvalues among the remaining eigenvalues.

Value

A list with elements eigval, len.block, eigvec, and chains. These may be easily merged with the output from make_mcev.

Note

This function covers all possibilities but has not been tested properly in the case when non-structural 0 eigenvalues are present.

TODO: Prepare meaningful examples and tests.

2015-12-27: dropped argument F0bot. The old signature was mc_0chains(dim, mo, mo.col, vec0, F0bot = NULL, flagtriang = TRUE).

Author(s)

Georgi N. Boshnakov

See Also

make_mcev

Examples

mc_0chains(6, 4, 1)
mc_0chains(6, 4, 2)
mc_0chains(6, 4, 3)
mc_0chains(6, 4, 4)

mc_0chains(6, 2, 2)
mc_0chains(6, 2, 4)
mc_0chains(6, 2, 3)
mc_0chains(6, 2, 3, list(matrix(1,              ncol=1, nrow=6)))
mc_0chains(6, 2, 3, list(matrix(c(1,0,0,0,0,0), ncol=1, nrow=6)))
mc_0chains(6, 2, 3, list(matrix(c(0,1,0,0,0,0), ncol=1, nrow=6)))
mc_0chains(6, 2, 3, list(matrix(c(0,0,0,0,0,1), ncol=1, nrow=6)))
mc_0chains(6, 2, 3, list(matrix(c(0,0,0,0,1,0), ncol=1, nrow=6)))

mc_0chains(2, 2, 2,list(matrix(c(0,1),ncol=1,nrow=2)))

mc_0chains(12,4,2)

mcompanion documentation built on Sept. 22, 2023, 5:12 p.m.