only for diagonalizable 2x2 matrices

squaremat.lua

function m_pow(matrix, n)
	local tr = matrix[1][1]+matrix[2][2] -- trace of matrix
	local s = math.sqrt((matrix[1][1]-matrix[2][2])^2+4*matrix[1][2]*matrix[2][1]) -- sqrt(tr^2 - 4*determinant)

	local r1 = 0.5*(tr+s) --1st eigenvalue
	local r2 = tr-r1 --2nd eigenvalue

	local r1n = r1^n
	local r2n = r2^n

	local b2 = (r1n-r2n)/s
	local b1 = (r1*r2n - r2*r1n)/s

	return {{b2*matrix[1][1]+b1, b2*matrix[1][2]}, {b2*matrix[2][1], b2*matrix[2][2]+b1}}

end

function print_matrix(matrix)
	print(unpack(matrix[1]))
	print(unpack(matrix[2]))
end

matrix = {{1,2},
		  {3,4}}
print_matrix(m_pow(matrix, 400))