Numpy matrix multiplication with array of matrices

Question

I have several numpy arrays that I would like to multiply (using dot, so matrix multiplication). I'd like to put them all into a numpy array, but I can't figure out how to do it.

E.g.

a = np.random.randn((10,2,2))
b = np.random.randn((10,2))

So I have 10 2x2 matrices (a) and 10 2x1 matrices (b). What I could do is this:

c = np.zeros((10,2))
for i in range(10):
  c[i] = np.dot(a[i,:,:],b[i,:])

You get the idea.

But I feel like there's a usage of dot or tensordot or something that would do this in one line really easily. I just can't make sense of the dot and tensordot functions for >2 dimensions like this.

unutbu · Accepted Answer · 2014-12-09 20:09:24Z

3

You could use np.einsum:

c = np.einsum('ijk,ik->ij', a, b)

einsum performs a sum of products. Since matrix multiplication is a sum of products, any matrix multiplication can be expressed using einsum. It is based on Einstein summation notation.

The first argument to einsum, ijk,ik->ij is a string of subscripts. ijk declares that a has three axes which are denoted by i, j, and k.

ik, similarly, declares that the axes of b will be denoted i and k.

When the subscripts repeat, those axes are locked together for purposes of summation. The part of the subscript that follows the -> shows the axes which will remain after summation.

Since the k appears on the left (of the ->) but disappears on the right, there is summation over k. It means that the sum

c_ij = sum over k ( a_ijk * b_ik )

should be computed. Since this sum can be computed for each i and j, the result is an array with subscripts i and j.

edited Dec 9, 2014 at 20:09

answered Dec 9, 2014 at 19:52

unutbu

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1 Comment

JoshG79 Over a year ago

That's great. That works. Would you mind writing just a few sentences about what the first argument means for those of us who don't use this notation?

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