Dot "power" of a matrix

Question

By analogy with the matrix product is there a name for the matrix "power" operation defined by $$y_i = \prod_j x_j^{a_{ij}}?$$ For example: $$\left( \begin{array}{lll} x_1 & x_2 & x_3\end{array}\right) \star \left( \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) = \left( \begin{array}{lll} x_1^{a_{11}}x_2^{a_{12}}x_3^{a_{13}} & x_2^{a_{21}}x_2^{a_{22}}x_3^{a_{23}} & x_3^{a_{31}}x_3^{a_{32}}x_3^{a_{33}} \end{array}\right)$$ I have come across this application in dimensional analysis where $\mathbf{x}$ is a vector of physical constants in one unit system and $\mathbf{A}$ the matrix of corresponding powers of dimensions.

Answer 1

Not normally given its own name, because it isn't really useful.

But you could take:

$$\left( \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) \left( \begin{array}{l} \log x_1\\ \log x_2\\ \log x_3\end{array}\right) = \begin{pmatrix}\log y_1\\ \log y_2\\ \log y_3\end{pmatrix}$$

using normal matrix multiplication, and get the same result.

So, it's a special case of the normal multiplication, and it doesn't occur enough to really warrant its own notation or name. I suppose you could call it "logarithmic linear transformation," but it's just likely to confuse people.

The altered question is the same, only with the transpose of what I wrote:

$$\left( \begin{array}{lll} \log x_1 &\log x_2 & \log x_3\end{array}\right) \left( \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) = \left( \begin{array}{lll} \log y_1 & \log y_2 & \log y_3 \end{array}\right)$$