Computing many quadratic forms in numpy

Suppose I have a collection of square matrices Sigma1, Sigma2, ... . These are stored in a numpy array Sigma such that Sigma[0, :, :] = Sigma1, Sigma[1, :, :] = Sigma2, etc.

I have a corresponding set of vectors v1, v2, ..., stored in a matrix v such that v[0, :] = v1, v[1, :] = v2, etc.

I am trying to produce a vector a such that a[0] is v1.T @ Sigma1 @ v1 (a scalar value), a[1] is v2.T @ Sigma2 @ v2, etc.

I'm a bit bewildered by the documentation for einsum tensordot and the like, which seem promising but which I have been unable to coerce into giving me the desired answer.

For example:

Sigma = np.empty(4, 3, 3)
for i in range(4):
    Z = np.random.randn(10, 3)
    Sigma[i, :, :] = Z.T @ Z

y = np.random.randn(4, 3)

I want a function foo(Sigma y) that returns a vector equivalent to

bar = np.empty(4)
for i in range 4:
    bar[i] = y[i, :].T @ Sigma[i, :, :] @ y[i, :]
return bar

source https://stackoverflow.com/questions/70385907/computing-many-quadratic-forms-in-numpy

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