Hadamard product - Knowlpedia

For two matrices $A,B$ of the same shape, the Hadamard product is their entrywise product

(A\odot B)_{ij}=A_{ij}B_{ij}.

For vectors, $(a\odot b)_i=a_i b_i$ . This is a different operation from ordinary matrix multiplication, which sums over an intermediate index.

Coordinate dependence

Entrywise multiplication uses the chosen coordinates. For a fixed vector $a$ , the map $b\mapsto a\odot b$ is multiplication by the diagonal matrix with entries $a_i$ . Thus differentiating coordinatewise squares gives $D(y\mapsto y\odot y)_y[h]=2y\odot h$ over the real or complex numbers.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.