zeeshas901
New member
- Joined
- Feb 14, 2020
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Hello! Can anyone please assist to solve the following question, please? Thank you for your time and efforts.
Let [MATH]\boldsymbol{u}=[u_{1}~~u_{2}~~\dots~~u_{n}][/MATH] and [MATH]\boldsymbol{v}=[v_{1}~~v_{2}~~\dots~~v_{n}][/MATH] be two known row vectors. If a square matrix [MATH]\boldsymbol{P}_{(n\times n)}=[p_{ij}][/MATH] is not a function of either [MATH]\boldsymbol{u}[/MATH] or [MATH]\boldsymbol{v}[/MATH] and the sum of each row of [MATH]\boldsymbol{P}[/MATH] is 1, where [MATH]p_{ij}>0[/MATH], then for which matrix [MATH]\boldsymbol{P}[/MATH] the following functional relation holds ([MATH]\boldsymbol{P}[/MATH] should not be identity matrix):
[MATH]f(\boldsymbol{uP},\boldsymbol{vP})=f(\boldsymbol{u},\boldsymbol{v}),[/MATH]
where the function [MATH]f(\cdot,\cdot)[/MATH] is defined as:
[MATH]f(\boldsymbol{x},\boldsymbol{y})=\sum\limits_{i=1}^{n}|y_{i}-x_{i}|.[/MATH]
Let [MATH]\boldsymbol{u}=[u_{1}~~u_{2}~~\dots~~u_{n}][/MATH] and [MATH]\boldsymbol{v}=[v_{1}~~v_{2}~~\dots~~v_{n}][/MATH] be two known row vectors. If a square matrix [MATH]\boldsymbol{P}_{(n\times n)}=[p_{ij}][/MATH] is not a function of either [MATH]\boldsymbol{u}[/MATH] or [MATH]\boldsymbol{v}[/MATH] and the sum of each row of [MATH]\boldsymbol{P}[/MATH] is 1, where [MATH]p_{ij}>0[/MATH], then for which matrix [MATH]\boldsymbol{P}[/MATH] the following functional relation holds ([MATH]\boldsymbol{P}[/MATH] should not be identity matrix):
[MATH]f(\boldsymbol{uP},\boldsymbol{vP})=f(\boldsymbol{u},\boldsymbol{v}),[/MATH]
where the function [MATH]f(\cdot,\cdot)[/MATH] is defined as:
[MATH]f(\boldsymbol{x},\boldsymbol{y})=\sum\limits_{i=1}^{n}|y_{i}-x_{i}|.[/MATH]