Maximizing Tr(A) & Unique Solution of Matrix A w/ Infinite Solutions

[zeeshas901]

Hello!

I am new here, and I need (urgent) help regarding the following question:

Let $\boldsymbol{A}_{(n\times n)}=[a_{ij}]$ be a square matrix such that the sum of each row is 1 and $a_{ij}\ge0$$(i=1,2,\dots,n~\text{and}~j=1,2,\dots,n)$ are unknown. Suppose that $\boldsymbol{b}_{1}=[b_{11}~b_{12}\dots b_{1n}]$ and $\boldsymbol{b}_{2}=[b_{21}~b_{22}\dots b_{2n}]$ are known row vectors of proportions such that $$\boldsymbol{b}_{1}\boldsymbol{A}_{(n\times n)}=\boldsymbol{b}_{2},$$ where $\boldsymbol{b}_{1}\boldsymbol{1}_{n}=1$, $\boldsymbol{b}_{2}\boldsymbol{1}_{n}=1$ and $\boldsymbol{1}^{T}_{n}=[1~1\dots1]$.

I know that there are infinite solutions for $\boldsymbol{A}$. However, I do not have any idea about the following two questions:

(i) how to optimize $\boldsymbol{A}$ such that trace$(\boldsymbol{A})$ is maximized (and each $a_{ij}$ may be expressed in terms of known quantities of the vectors $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{2}$ if possible) subject to the sum of each row of $\boldsymbol{A}$ is 1
and
(ii) under which condition(s) a unique $\boldsymbol{A}$ exists?Thank you!

 

[jvicens]

Hello there,

Thank you for reaching out for help with your question. Let me try to break down the problem and provide some suggestions for solving it.

Firstly, let's define the variables and constraints in the problem:

- $\boldsymbol{A}$ is a square matrix of size $n\times n$
- $a_{ij}$ are unknown elements of the matrix, with $i,j=1,2,\dots,n$
- The sum of each row of $\boldsymbol{A}$ is equal to 1, i.e. $\sum_{j=1}^{n}a_{ij}=1$
- $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{2}$ are known row vectors of size $1\times n$
- $\boldsymbol{b}_{1}\boldsymbol{A}=\boldsymbol{b}_{2}$, with the additional constraints that $\boldsymbol{b}_{1}\boldsymbol{1}_{n}=1$ and $\boldsymbol{b}_{2}\boldsymbol{1}_{n}=1$

Now, let's consider the first question, i.e. how to optimize $\boldsymbol{A}$ such that trace$(\boldsymbol{A})$ is maximized. The trace of a matrix is the sum of its diagonal elements, so maximizing the trace of $\boldsymbol{A}$ is equivalent to maximizing the sum of its diagonal elements. This can be achieved by setting $a_{ii}=1$ for all $i=1,2,\dots,n$. This satisfies the constraint that the sum of each row is 1 and also maximizes the trace of $\boldsymbol{A}$. However, this solution may not be unique, and there may be other solutions that also satisfy the constraints and have the same maximum trace.

As for expressing each $a_{ij}$ in terms of known quantities of $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{2}$, this may be possible depending on the specific values of $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{2}$. For example, if $\boldsymbol{b}_{1}=[1~0~0\dots0]$ and $\boldsymbol{b}_{2}=[0~1~0\dots0]$, then we can set $a_{11}=0$ and $a_{22}=0$, and the remaining elements can be expressed in