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Can the product(element to element) of two vector $\vec{a}$ and $\vec{b}$ with the same dimension be written as the usual vector(or matrix) product?

Say $\vec{a}=[2,3],\vec{b}=[4,5]$ and $\vec{a}~\text{product(element to element)}~\vec{b}=[8,15]$

Thank you!

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if you mean dot product ,then it is just scalar not vector,in your case a*b=23 – dato datuashvili Jan 11 '12 at 19:56
Yes, I know. I mean I can add some other constant matrix like identity matrix, or other constant vector like $e_j$, anyway, I just want to get rid off that kind of unusual product. – breezeintopl Jan 11 '12 at 20:01
up vote 3 down vote accepted

No, not if you mean standard matrix multiplication. The matrix product of two $n\times 1$ or $1\times n$ matrices does not exist. Even if you allow taking the transpose, the product of a $1\times n$ matrix and a $n\times 1$ matrix is $1\times 1$ while the product of a $n\times 1$ matrix and a $1\times n$ matrix is $n\times n$. None of these are vectors, so this can't be what you are looking for.

However, as Andreas pointed out, there is a name for the operation you are talking about, the Hadamard product, and it generalizes to all matrices.

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Ok, thank you so much! I understand. – breezeintopl Jan 11 '12 at 20:08

For matrices there is the Hadamard product with $ (A \circ B)_{i,j} = (A)_{i,j} \cdot (B)_{i,j}. $ So think of vectors as $1-\text{by}-2$ matrices.

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