Arithmetic Operations¶

Sparse matrices support efficient arithmetic operations.

Addition and Subtraction¶

1. Sparse + Sparse¶

from scipy import sparse

def main():
    A = sparse.csr_matrix([[1, 0], [0, 2]])
    B = sparse.csr_matrix([[0, 3], [4, 0]])

    C = A + B

    print("A + B =")
    print(C.toarray())

if __name__ == "__main__":
    main()

2. Sparse + Scalar¶

from scipy import sparse

def main():
    A = sparse.csr_matrix([[1, 0], [0, 2]])

    # Note: adds to ALL entries (converts to dense pattern)
    B = A + 1

    print("A + 1 =")
    print(B.toarray())

if __name__ == "__main__":
    main()

Scalar Multiplication¶

1. Scale Matrix¶

from scipy import sparse

def main():
    A = sparse.csr_matrix([[1, 0, 2], [0, 3, 0]])

    B = 2 * A
    C = A * 0.5

    print("2 * A =")
    print(B.toarray())

if __name__ == "__main__":
    main()

Matrix-Vector Product¶

1. Sparse @ Dense¶

import numpy as np
from scipy import sparse

def main():
    A = sparse.csr_matrix([[1, 0, 2],
                           [0, 3, 0],
                           [4, 0, 5]])
    x = np.array([1, 2, 3])

    y = A @ x

    print(f"A @ x = {y}")

if __name__ == "__main__":
    main()

2. Performance¶

import numpy as np
from scipy import sparse
import time

def main():
    n = 10000
    A = sparse.random(n, n, density=0.001, format='csr')
    x = np.random.randn(n)

    start = time.perf_counter()
    for _ in range(100):
        y = A @ x
    elapsed = time.perf_counter() - start

    print(f"100 sparse matvecs: {elapsed:.4f} sec")

if __name__ == "__main__":
    main()

Element-wise Operations¶

1. Hadamard Product¶

from scipy import sparse

def main():
    A = sparse.csr_matrix([[1, 2], [3, 4]])
    B = sparse.csr_matrix([[1, 0], [0, 1]])

    # Element-wise multiplication
    C = A.multiply(B)

    print("A ⊙ B =")
    print(C.toarray())

if __name__ == "__main__":
    main()

2. Power¶

from scipy import sparse

def main():
    A = sparse.csr_matrix([[1, 2], [3, 4]])

    # Element-wise power
    B = A.power(2)

    print("A.^2 =")
    print(B.toarray())

if __name__ == "__main__":
    main()

Summary¶

Operation	Method	Preserves Sparsity
A + B	`A + B`	Yes
A + scalar	`A + c`	No (fills matrix)
c * A	`c * A`	Yes
A @ x	`A @ x`	N/A (returns dense)
A ⊙ B	`A.multiply(B)`	Yes