0
$\begingroup$

Consider the following nonlinear matrix equation:

$B=PX^{−1}AX$

where $B$ and $P$ are a $1\times n$ row vector and $A$ is a $n\times n$ matrix which are all strictly positive, and $X=diag(x_1,...,x_n)$ is a diagonal matrix with $x_i>0$ for all $i$'s. $X$ is the unknown while the rest are all taken as given.

I'm looking for the general solution for $X$ and for its existence and uniqueness conditions. But it seems that the general solution does not exist. If so, how can we formally prove that?

I read this post but it had no answer for me.

Improve this question
$\endgroup$
6
  • 5
    $\begingroup$ If you multiply on the right by $X^{-1}$, then you obtain a linear equation in the $x_i^{-1}$. $\endgroup$
    – Anthony Quas
    Commented Jun 25, 2019 at 4:33
  • $\begingroup$ Thanks, Anthony! But I think it would still be nonlinear equation in $x_i^{-1}$, no? $\endgroup$
    – ppp
    Commented Jun 25, 2019 at 5:18
  • 1
    $\begingroup$ If $X$ is a solution then $\alpha X$ also is a solution, for each $\alpha > 0$. So, there is no uniqueness condition, in general. $\endgroup$
    – Mahdi - Free Palestine
    Commented Jun 25, 2019 at 5:54
  • 2
    $\begingroup$ No. If you call $y_i=x_i^{-1}$, then both sides are linear expressions in the $y_i$ (try writing out the 2 sides). $\endgroup$
    – Anthony Quas
    Commented Jun 25, 2019 at 5:56
  • 1
    $\begingroup$ Following up on Anthony's comment, letting $Y = (x_1^{-1}, \ldots, x_n^{-1})$ as a row vector, the resulting equation is $Y(\hat{B}-\hat{P}A)=0$, where $\hat{B}$ and $\hat{P}$ are now diagonal matrices with the elements of $B$ and $P$ on the diagonal, respectively. $\endgroup$
    – Igor Khavkine
    Commented Jun 25, 2019 at 13:46

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .