Multiplying array of vectors by matrix without for loop

Question

I have a 2 x 2 numpy.array() matrix, and an array N x 2 X, containing N 2-dimensional vectors.

I want to multiply each vector in X by the 2 x 2 matrix. Below I use a for loop, but I am sure there is a faster way. Please, could someone show me what it is? I assume there is a way using a numpy function.

# the matrix I want to multiply X by
matrix = np.array([[0, 1], [-1, 0]]) 

# initialize empty solution
Y = np.empty((N, 2))

# loop over each vector in X and create a new vector Y with the result
for i in range(0, N):
    Y[i] = np.dot(matrix, X[i])

For example, these arrays:

matrix = np.array([
    [0,  1],
    [0, -1]
])

X = np.array([
    [0, 0],
    [1, 1],
    [2, 2]
])

Should result in:

Y = np.array([
    [0,  0],
    [1, -1],
    [2, -2]
])

Asked By: johnny rotten

||

Source

Answer 1

You are doing some kind of matrix multiplication with (2,2) matrix and each (2,1) X line.
You need to make all your vectors the same dimension to directly calculate this. Add a dimension with None and directly calculate Y like this :

matrix = np.array([[3, 1], [-1, 0.1]]) 
N = 10
Y = np.empty((N, 2))
X =np.ones((N,2))
X[0][0] = 2
X[5][1] = 3
# loop over each vector in X and create a new vector Y with the result
for i in range(0, N):
    Y[i] = np.dot(matrix, X[i])

Ydirect = matrix[None,:] @ X[:,:,None]
print(Y)
print(Ydirect[:,:,0])

Answered By: Adrien Mau

Answer 2

You can vectorize Adrien’s result and remove the for loop, which will optimize performance, especially as the matrices get bigger.

matrix = np.array([[3, 1], [-1, 0.1]])

N = 10

X = np.ones((N, 2))
X[0][0] = 2
X[5][1] = 3

# calculate dot product using @ operator
Y = matrix @ X.T


print(Y)

Answered By: Michael Varney

Answer 3

One-liner is (matrix @ X.T).T

Just transpose your X, to get your vectors in columns. Then matrix @ X.T or (np.dot(matrix, X.T) if you prefer this solution, but now that @ notation exists, why not using it) is a matrix made of columns of matrix times X[i]. Just transpose back the result if you need Y to be made of lines of results

matrix = np.array([[0, 1], [-1, 0]]) 
X = np.array([[1,2],[3,4],[5,6]])
Y = (matrix @ X.T).T

Y is

array([[ 2, -1],
       [ 4, -3],
       [ 6, -5]])

As expected, I guess.

In detail:
X is

array([[1, 2],
       [3, 4],
       [5, 6]])

so X.T is

array([[1, 3, 5],
       [2, 4, 6]])

So, you can multiply your 2×2 matrix by this 2×3 matrix, and the result will be a 2×3 matrix whose columns are the result of multiplication of matrix by the column of this. matrix @ X.T is

array([[ 2,  4,  6],
       [-1, -3, -5]])

And transposing back this gives the already given result.

So, tl;dr: one-liner answer is (matrix @ X.T).T

Answered By: chrslg

Multiplying array of vectors by matrix without for loop

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