# Entrywise Conversion from Matrix of Integers to Matrix of Reciprocals

Suppose I have a matrix where $x \in \mathbb{Z}$

$A = \begin{bmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{d1} & x_{d2} & x_{d3} & \dots & x_{dn} \end{bmatrix}$

what is the best way to arrive at the following matrix:

$B = \begin{bmatrix} \frac{1}{x_{11}} & \frac{1}{x_{12}} & \frac{1}{x_{13}} & \dots & \frac{1}{x_{1n}} \\ \frac{1}{x_{21}} & \frac{1}{x_{22}} & \frac{1}{x_{23}} & \dots & \frac{1}{x_{2n}} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{1}{x_{d1}} & \frac{1}{x_{d2}} & \frac{1}{x_{d3}} & \dots & \frac{1}{x_{dn}} \end{bmatrix}$

• Find the reciprocal of each element. – Kenny Lau Aug 2 '16 at 8:22
• yes...how would you write this? – TsTeaTime Aug 2 '16 at 8:22
• "Find the reciprocal of each element." – Kenny Lau Aug 2 '16 at 8:22
• I don't suppose i can write $1/A$ – TsTeaTime Aug 2 '16 at 8:23
• "$B$ is the matrix of the reciprocal of each element in $A$" – Kenny Lau Aug 2 '16 at 8:24

If $A, B \in \mathbb K^{n \times m}$ then the Schurproduct $A \ast B$ is defined as $(A \ast B)_{i,j} = (A)_{i,j} (B)_{i,j}$, i.e. the multiplication of the entries.
"Let $B$ the inverse of $A$ respecting the Schur product."
• Does $\mathbb K^{n \times m}$ represent anything in particular? i am not able to find this under the list of Number Sets? – TsTeaTime Aug 2 '16 at 8:49
• In my notation you have $\mathbb K \in \{\mathbb R, \mathbb C\}$ and $\mathbb K^{n \times m}$ denotes the space of matrices with $n$ rows and $m$ columns with entries in $\mathbb K$. Hope that makes it a little clearer :) – Yaddle Aug 2 '16 at 9:19