For which matrix, the following functional relation holds?

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zeeshas901

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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]
 
zeeshas901 said:
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]
Click to expand...
Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this assignment.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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]
 
Try to work it out for n=2. See what you learn from that.
 
I tried working this out for n=2 and that P= identity matrix.

Maybe knowing what P is not a function of u and v means might help??
 
Subhotosh Khan said:
Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this assignment.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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]
Click to expand...

I tried to solve like this:
Let [MATH]\boldsymbol{u}=[u_{i=1}~~u_{2}][/MATH], [MATH]\boldsymbol{v}=[v_{i=1}~~v_{2}][/MATH] and [MATH]\boldsymbol{P}= \begin{bmatrix} p_{11} & p_{12}\\ p_{21} & p_{22} \end{bmatrix}. [/MATH]Thus,
[MATH]f(\boldsymbol{u},\boldsymbol{v})=\sum\limits_{i=1}^{2}|v_{i}-u_{i|}[/MATH]and
[MATH]f(\boldsymbol{uP},\boldsymbol{vP})=\sum\limits_{i=1}^{2}(\sum\limits_{j=1}^{2}p_{ij|}v_{i}-u_{i}|)[/MATH].
If [MATH]f(\boldsymbol{u},\boldsymbol{v})=f(\boldsymbol{uP},\boldsymbol{vP}),[/MATH] then [MATH]\boldsymbol{P}[/MATH] should be identity matrix. But in the question, it is stated that [MATH]\boldsymbol{P}[/MATH] should not be identity.

Now I revise my question like `for what class of [MATH]g(\boldsymbol{u})[/MATH] and [MATH]g(\boldsymbol{v})[/MATH]' the following holds.

[MATH]f(g(\boldsymbol{u}),g(\boldsymbol{v}))=f(\boldsymbol{u},\boldsymbol{v}),[/MATH]where [MATH]g(\boldsymbol{u})[/MATH] is a functions of [MATH]\boldsymbol{u}[/MATH] and [MATH]\boldsymbol{P}[/MATH] and [MATH]g(\boldsymbol{v})[/MATH] is a functions of [MATH]\boldsymbol{v}[/MATH] and [MATH]\boldsymbol{P}[/MATH]. Further [MATH]f(\cdot,\cdot)[/MATH] is the first norm as defined in the question.
 
Last edited:
Jomo said:
Try to work it out for n=2. See what you learn from that.
Click to expand...

I tried to solve like this:
Let [MATH]\boldsymbol{u}=[u_{i=1}~~u_{2}][/MATH], [MATH]\boldsymbol{v}=[v_{i=1}~~v_{2}][/MATH] and [MATH]\boldsymbol{P}= \begin{bmatrix} p_{11} & p_{12}\\ p_{21} & p_{22} \end{bmatrix}. [/MATH]Thus,
[MATH]f(\boldsymbol{u},\boldsymbol{v})=\sum\limits_{i=1}^{2}|v_{i}-u_{i|}[/MATH]and
[MATH]f(\boldsymbol{uP},\boldsymbol{vP})=\sum\limits_{i=1}^{2}(\sum\limits_{j=1}^{2}p_{ij|}v_{i}-u_{i}|)[/MATH].
If [MATH]f(\boldsymbol{u},\boldsymbol{v})=f(\boldsymbol{uP},\boldsymbol{vP}),[/MATH] then [MATH]\boldsymbol{P}[/MATH] should be identity matrix. But in the question, it is stated that [MATH]\boldsymbol{P}[/MATH] should not be identity.

Now I revise my question like `for what class of [MATH]g(\boldsymbol{u})[/MATH] and [MATH]g(\boldsymbol{v})[/MATH]' the following holds.

[MATH]f(g(\boldsymbol{u}),g(\boldsymbol{v}))=f(\boldsymbol{u},\boldsymbol{v}),[/MATH]where [MATH]g(\boldsymbol{u})[/MATH] is a functions of [MATH]\boldsymbol{u}[/MATH] and [MATH]\boldsymbol{P}[/MATH] and [MATH]g(\boldsymbol{v})[/MATH] is a functions of [MATH]\boldsymbol{v}[/MATH] and [MATH]\boldsymbol{P}[/MATH]. Further, the function [MATH]f(\cdot,\cdot)[/MATH] is the first normal as defined in the question.
 
Jomo said:
I thought that but also thought that it was strange to state.
Click to expand...
Maybe it is strange. But I am trying to find out those g(u) and g(v) for which the above functional relation, given in the question, holds.
 
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