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I don't know the name of this type of matrix multiplication. In this type, we multiply each same indexes of matrices and generate our new matrix. Please help me find out.

Normal matrix multiplication is like this: $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} * \begin{bmatrix} x & y \\ w & z \end{bmatrix} = \begin{bmatrix} ax + bw & ay +bz \\ cx + dw & cy + dz \end{bmatrix} $$

But what is the name of this multiplication? $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} * \begin{bmatrix} x & y \\ w & z \end{bmatrix} = \begin{bmatrix} ax & by \\ cw & dz \end{bmatrix} $$

Thanks in advance.

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    $\begingroup$ We'd probably call it "component-wise," but it's pretty useless. (Which is why it doesn't have a "real" name.) $\endgroup$
    – Randall
    Oct 12, 2018 at 18:33
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    $\begingroup$ It is Hadamard's product. You can take a look at[Wikipedia] (en.wikipedia.org/wiki/Hadamard_product_(matrices)). $\endgroup$
    – Bernard
    Oct 12, 2018 at 18:35
  • $\begingroup$ thank you @Bernard $\endgroup$
    – Peyman
    Oct 12, 2018 at 18:41
  • $\begingroup$ It's actually pretty useful, which is why all popular numerical computation systems support it. $\endgroup$ Oct 12, 2018 at 18:49
  • $\begingroup$ Well, color me stunned. You learn something every day here. $\endgroup$
    – Randall
    Oct 12, 2018 at 18:56

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As mentioned above this is the element-/entry-/point-wise product and it is the most intuitive way of defining matrix multiplication! However, standard matrix multiplication (which should really be called "composition") has more interesting properties in linear algebra, namely if $$ f: \vec y \mapsto A\vec y,g: \vec x \mapsto B\vec x$$ then $$ f\circ g: \vec x \mapsto AB\vec x$$ In other words $f(g(\vec x)) = AB\vec x$ and $AB$ is the matrix representing the homomorphism (linear function) $f\circ g$

Note that for two vectors of same length $x,y$ we have: $$ xy^T = (x_i y_i) $$ which is a kind of entrywise/pairwise multiplication.

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