mc.0chain.complete: Extend or shrink eigenchains of 0 eigenvalues

In mcompanion: Objects and Methods for Multi-Companion Matrices

Extend or shrink eigenchains of 0 eigenvalues

Description

Takes a Jordan chain of the 0 eigenvalue of a multi-companion matrix and modifies it to be a Jordan chain of a larger or smaller multi-companion matrix.

Usage

mc.0chain.complete(dim, mo, chain, alt0)

Arguments

dim	the size of the new matrix, a number.
mo	the multi-companion order of the matrix.
chain	the chain from which the new chain is to be derived.
alt0	optional alternative initialisation for the free elements, see Details.

Details

If the number of rows of chain is larger than or equal to mo, then chain represents a Jordan chain of the source mc-matrix with the eigenvector is in the first column.

Otherwise (if nrow(chain) < mo) the number of rows is taken to be mo.col and the Jordan chain is that of the top mo.col x mo.col corner. In this case, the chain is extended first to a chain for the top left mo x mo corner. Argument F0bot allows this to be accomplished. It provides the [(mo.col+1):mo, 1:mo] block of the mc-matrix.

dim specifies the dimension of the modified matrix.

The number of elements in the new chain may be different from the original and the eigenvector may not be a shrunk version of the original eigenvector.

The new Jordan chain is returned as a matrix of dim rows and number of columns determined automatically.

When the new matrix is larger than the original, some of the elements of the last vector in the new chain are arbitrary. By default these elements are set to zero. Argument alt0 can be used to change this. It should be a vector of length dim - nrow(chain).

Value

a matrix

Author(s)

Georgi N. Boshnakov

Examples

x1 <- cbind(c(1,1), c(1,2))
j1 <- diag(c(0.8, 0.5))
m1 <- x1 %*% j1 %*% solve(x1)
bo1 <- rbind(c(0.5, 0.8), c(0.256, 0.512))

j1a <- diag(c(0, 0.5))
m1a <- x1 %*% j1a %*% solve(x1)
f1a <- cbind( rbind(m1a, bo1), 0, 0 )

mcompanion documentation built on May 29, 2024, 4:31 a.m.