A simple function to demonstrate calculating the power of a square symmetric matrix in terms of its eigenvalues and eigenvectors.
mpower(A, p, tol = sqrt(.Machine$double.eps))a square symmetric matrix
matrix power, not necessarily a positive integer
tolerance for determining if the matrix is symmetric
A raised to the power p: A^p
The matrix power p can be a fraction or other non-integer. For example, p=1/2 and
p=1/3 give a square-root and cube-root of the matrix.
Negative powers are also allowed. For example, p=-1 gives the inverse and p=-1/2
gives the inverse square-root.
The {%^%} operator in the expm package is far more efficient
C <- matrix(c(1,2,3,2,5,6,3,6,10), 3, 3) # nonsingular, symmetric
C
#> [,1] [,2] [,3]
#> [1,] 1 2 3
#> [2,] 2 5 6
#> [3,] 3 6 10
mpower(C, 2)
#> [,1] [,2] [,3]
#> [1,] 14 30 45
#> [2,] 30 65 96
#> [3,] 45 96 145
zapsmall(mpower(C, -1))
#> [,1] [,2] [,3]
#> [1,] 14 -2 -3
#> [2,] -2 1 0
#> [3,] -3 0 1
solve(C) # check
#> [,1] [,2] [,3]
#> [1,] 14 -2 -3
#> [2,] -2 1 0
#> [3,] -3 0 1