Characteristic Polynomial

Computes the characteristic polynomial (and the inverse of the matrix, if requested) using the Faddeew-Leverrier method.

charpoly(a, info = FALSE)

Arguments

a

quadratic matrix; size should not be much larger than 100.

info

logical; if true, the inverse matrix will also be reported.

Details

Computes the characteristic polynomial recursively. In the last step the determinant and the inverse matrix can be determined without any extra cost (if the matrix is not singular).

Value

Either the characteristic polynomial as numeric vector, or a list with components cp, the characteristic polynomial, det, the determinant, and inv, the inverse matrix, will be returned.

References

Hou, S.-H. (1998). Classroom Note: A Simple Proof of the Leverrier–Faddeev Characteristic Polynomial Algorithm, SIAM Review, 40(3), pp. 706–709.

Examples

a <- magic(5)
A <- charpoly(a, info = TRUE)
#> Error term: 0 
A$cp
#> [1]        1      -65     -625    40625    78000 -5070000
roots(A$cp)
#> [1]  65.00000 -21.27677  21.27677 -13.12628  13.12628
A$det
#> [1] 5070000
zapsmall(A$inv %*% a)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    0    0    0    0
#> [2,]    0    1    0    0    0
#> [3,]    0    0    1    0    0
#> [4,]    0    0    0    1    0
#> [5,]    0    0    0    0    1