Does this kind of matrix product exist?

[lets_resonate]

Suppose you have a matrix A:

$$\left(<br /> \begin{array}{ccc}<br /> a_{1,1} & a_{1,2} & a_{1,3} \\<br /> a_{2,1} & a_{2,2} & a_{2,3} \\<br /> a_{3,1} & a_{3,2} & a_{3,3}<br /> \end{array}<br /> \right)$$

And a matrix B:

$$\left(<br /> \begin{array}{ccc}<br /> b_{1,1} & b_{1,2} & b_{1,3} \\<br /> b_{2,1} & b_{2,2} & b_{2,3} \\<br /> b_{3,1} & b_{3,2} & b_{3,3}<br /> \end{array}<br /> \right)$$

I want a product $A \star B$ that would result in:

$$\left(<br /> \begin{array}{ccc}<br /> a_{1,1} b_{1,1} & a_{1,2} b_{1,2} & a_{1,3} b_{1,3} \\<br /> a_{2,1} b_{2,1} & a_{2,2} b_{2,2} & a_{2,3} b_{2,3} \\<br /> a_{3,1} b_{3,1} & a_{3,2} b_{3,2} & a_{3,3} b_{3,3}<br /> \end{array}<br /> \right)$$

Does such a product exist? What would be the name of it?

 

[Mute]

It certainly exists. For one thing, you just defined it! But regardless of that fact, this is a common product in MATLAB: v1.*v2, where v1 and v2 are vectors or matrices of the same size. I'm not sure if it has a specific name, probably because it's a special case of a more general operation, the tensor product.

I can define a rank-4 tensor with components

$$C^{ik}_{~~jl} = A^{i}_{~j}B^{k}_{~l}$$

I can then contract over the indices k and j to get the matrix you want:

$$M^{i}_{~l} = C^{ik}_{~~kl} = A^{i}_{~k}B^{k}_{~l}$$

In case you're unfamilar with index notation, we're basically specifying the components of the matrix (or, in general, higher dimensional objects called tensors) using indices to label the rows and columns (and the other higher dimensional directions of the object). Note that I've used "Einstein Notation", meaning a summation is implied: any time an index appears twice on a product or term it is being summed over. The tensor C I defined is a new object defined in terms of the components of two other objects. The "Contraction" I did, summing over elements of the C tensor, results in a lower-dimensional object, which in this case is the object you defined.

See: http://en.wikipedia.org/wiki/Einstein_notation , http://en.wikipedia.org/wiki/Tensor_product , http://en.wikipedia.org/wiki/Tensor

Note that the stuff on tensors may be a little more than you want or need for your purposes, depending on what they are.

 

[Hurkyl]

I can then contract over the indices k and j to get the matrix you want:

$$M^{i}_{~l} = C^{ik}_{~~kl} = A^{i}_{~k}B^{k}_{~l}$$

No, that is the ordinary product of two matrices.

 

[Hurkyl]

What would be the name of it?

My first guess is that someone using such a product would call it the "pointwise product", "componentwise product", or "elementwise product", if they bothered to name it at all.

 

[Päällikkö]

Hadamard product is the name I've seen used for it.

 

[matt grime]

I've seen it used in C* algebra related material, though I can't recall what it was used for.