Matrix Power

Compute arbitrary matrix powers \(A^p\) including fractional exponents.

linalg.fractional_matrix_power

1. Basic Usage

import numpy as np
from scipy import linalg

def main():
    A = np.array([[4, 0],
                  [0, 9]])

    # A^{0.5} = sqrt(A)
    A_half = linalg.fractional_matrix_power(A, 0.5)

    print("A =")
    print(A)
    print()
    print("A^{0.5} =")
    print(A_half)

if __name__ == "__main__":
    main()

2. Various Powers

import numpy as np
from scipy import linalg

def main():
    A = np.array([[2, 1],
                  [0, 3]])

    powers = [-1, -0.5, 0, 0.5, 1, 2]

    for p in powers:
        A_p = linalg.fractional_matrix_power(A, p)
        print(f"A^{p:4.1f} =")
        print(A_p.real.round(4))
        print()

if __name__ == "__main__":
    main()

3. Verify A^p @ A^q = A^{p+q}

import numpy as np
from scipy import linalg

def main():
    A = np.array([[2, 1],
                  [1, 2]])

    p, q = 0.3, 0.7

    A_p = linalg.fractional_matrix_power(A, p)
    A_q = linalg.fractional_matrix_power(A, q)
    A_pq = linalg.fractional_matrix_power(A, p + q)

    print(f"A^{p} @ A^{q} =")
    print((A_p @ A_q).real.round(6))
    print()
    print(f"A^{p+q} =")
    print(A_pq.real.round(6))

if __name__ == "__main__":
    main()

Special Cases

1. Integer Powers

import numpy as np
from scipy import linalg

def main():
    A = np.array([[1, 2],
                  [3, 4]])

    # Integer power via fractional_matrix_power
    A_3 = linalg.fractional_matrix_power(A, 3)

    # Direct computation
    A_3_direct = A @ A @ A

    print("A³ via fractional_matrix_power:")
    print(A_3.real.round(6))
    print()
    print("A³ direct:")
    print(A_3_direct)

if __name__ == "__main__":
    main()

2. Negative Powers

import numpy as np
from scipy import linalg

def main():
    A = np.array([[2, 1],
                  [1, 2]])

    # A^{-1}
    A_inv = linalg.fractional_matrix_power(A, -1)

    # Verify
    print("A^{-1} @ A =")
    print((A_inv @ A).real.round(10))

if __name__ == "__main__":
    main()

3. Zero Power

import numpy as np
from scipy import linalg

def main():
    A = np.array([[1, 2],
                  [3, 4]])

    # A^0 = I
    A_0 = linalg.fractional_matrix_power(A, 0)

    print("A^0 =")
    print(A_0.real.round(10))

if __name__ == "__main__":
    main()

Applications

1. Matrix Interpolation

import numpy as np
from scipy import linalg

def main():
    A = np.array([[1, 0],
                  [0, 1]])
    B = np.array([[4, 0],
                  [0, 9]])

    # Interpolate: C(t) = A^{1-t} @ B^t
    for t in [0, 0.25, 0.5, 0.75, 1.0]:
        A_1mt = linalg.fractional_matrix_power(A, 1-t)
        B_t = linalg.fractional_matrix_power(B, t)
        C_t = A_1mt @ B_t
        print(f"t={t}: diag = {np.diag(C_t.real).round(2)}")

if __name__ == "__main__":
    main()

2. Diffusion Process

import numpy as np
from scipy import linalg

def main():
    # Transition matrix
    P = np.array([[0.7, 0.3],
                  [0.4, 0.6]])

    # Fractional steps
    for t in [0.5, 1, 2, 10]:
        P_t = linalg.fractional_matrix_power(P, t)
        print(f"P^{t} =")
        print(P_t.real.round(4))
        print()

if __name__ == "__main__":
    main()

Summary

Function	Description
linalg.fractional_matrix_power(A, p)	Compute \(A^p\)

Key: Works for any real \(p\). Uses principal branch for non-integer powers.