0
$\begingroup$

Suppose I have a matrix $A$ with entries $A_{ij}$ and a vector $u$ with entries $u_i$, and that I want to express matrix $C$ with entries $C_{ij} = A_{ij}u_i$ as a product of matrices or vectors and matrices.

Example:

$A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}$ $u=\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}$

$C=\begin{bmatrix}u_{1}a_{11}&u_{1}a_{12}\\u_{2}a_{21}&u_{2}a_{22}\end{bmatrix}$

Is there a standard notation similar to that of the Hadamard product ⊙ to write this?

If not, is there a standard notation to express that a diagonal matrix M is generated using the entries of vector $u$ as the values for the diagonal, like the MATLAB and Numpy diag(u) function

$M=\begin{bmatrix}u_{1}& 0\\0&u_{2}\end{bmatrix}$

so that I can do $C = MA$?

$\endgroup$
1
$\begingroup$

Actually I suggest to be more clear in what you mean by "multiplying a matrix $A$ by a vector $u$". The product will be a vector! Not a matrix!

Remember that, given a field $K$, an object $A\in K^{m\times n}$ can be multiplied canonically only by a column vector $u \in K^{n\times 1}$ and the product $Au \in K^{m\times 1}$.

In any case, if I correctly uderstood your question, you're looking for an operator that extends this notation. Then I suggest you to look to two notations:

  1. Einstein Notation for the repeated indices: When you are for example in $\mathbb{R}^{3}$, you will have that the indices can range over the set $i={1, 2, 3}$, $$y=\sum_{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}$$ is simplified by the convention to: $$y=c_{i}x^{i}$$ The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors.
  2. Tensor Product (generalization of Matric product in higher dimension) denoted by $\otimes$ , from ordered pairs in the Cartesian product $V\times W$ into $V\otimes W$, in a way that generalizes the outer product.
$\endgroup$
  • $\begingroup$ I don't know how I could be clearer, I expressed exactly what the entries of C should be given the entries of A and u... I also said it was somewhat of an element wise multiplication, and I mentioned the Hadamard product to give an idea of what I mean. $\endgroup$ – nayriz Jun 16 '17 at 16:46
  • $\begingroup$ Do you remember that $C$ is a vector and Not a Matrix? Using two sub indices does not make any sense! Otherwise, if you want to denote the $i$-th element of $C$ and you want to express it as the combination of the element of $A$ and $u$, it can be easily done with the Einstein notation. For example $C_i = A_{ij}u^j$ $\endgroup$ – Erik Pillon Jun 16 '17 at 16:53
  • $\begingroup$ C is a matrix, please see edits. $\endgroup$ – nayriz Jun 16 '17 at 16:59
  • $\begingroup$ @nayriz I bag your pardon... actually I think that the notation you wrote is the most used $\endgroup$ – Erik Pillon Jun 16 '17 at 17:06
  • $\begingroup$ No problem! How about a notation to use a vector to create a diagonal matrix? That'd do the trick... $\endgroup$ – nayriz Jun 16 '17 at 17:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.