mvinfluence (version 0.9.0)

mpower: General Matrix Power

Description

Calculates the n-th power of a square matrix, where n can be a positive or negative integer or a fractional power.

Usage

mpower(A, n)
A %^% n

Value

Returns the matrix \(A^n\)

Arguments

A

A square matrix. Must also be symmetric for non-integer powers.

n

matrix power

Author

Michael Friendly

Details

If n<0, the method is applied to \(A^{-1}\). When n is an integer, the function uses the Russian peasant method, or repeated squaring for efficiency. Otherwise, it uses the spectral decomposition of A, \(\mathbf{A}^n = \mathbf{V} \mathbf{D}^n \mathbf{V}^{T}\) requiring a symmetric matrix.

References

https://en.wikipedia.org/wiki/Exponentiation_by_squaring

See Also

Packages corpcor and expm define similar functions.

Examples

M <- matrix(sample(1:9), 3,3)
mpower(M,2)
mpower(M,4)

# make a symmetric matrix
MM <- crossprod(M)
mpower(MM, -1)
Mhalf <- mpower(MM, 1/2)
all.equal(MM, Mhalf %*% Mhalf)