Let $b = (b_1,\ldots,b_n)\in \mathbb{R}^n$ such that $\forall i \in \left\{ 1,\ldots, n \right\}$ we have $b_i > 0$ and $\forall i \in \left\{1,\ldots,n-1 \right\}$ we have $b_i > b_{i+1}$. Let also $x \in \mathbb{R}^n$ such that
$$Ax = b$$
where $A \in \mathbb{R}^{n \times n}$ is an invertible...
Your definition is perfect. I want to provide a formal definition and give literature reference.
My guess is since elements of a vector can be arbitrarily interchanged, any vector can become monotonic and thus there is no formal concept of monotonic vector. And the transformation may not result in a monotonic vector. I have found out Monotone Operators and Monotone Mapping. Definition for Monotone Mapping is given in Variational Analysis Chapter 12. Not sure if it is relevant to your question.
Well if $A$ is diagonal with positive entries the monotonic behaviour won't change, indeed I was asking for condition / constraints on the matrix $A$ to preserve such behaviour. I came across monotone operators and I thought they might be related, but I haven't investigated this since. How come you're looking into this?
As part of my dissertation, I want to prove that the transformation that I have derived from a Markov chain gives a vector that is monotone when the input vector is monotone. There isn't any definition I have come across where for monotone mapping in explained in simple english or by using an algebraic or numerical example. Also the only way to prove monotonicity is to use partial derivatives or to prove using the definition of monotone mapping. Sadly I lack knowledge on handling derivatives related to system of linear equations that are used in stationary probabilities of Markov chain.
I'm actually still interested in that problem I'd like to know if there're conditions
The interpretation I'd give to the monotone operator is that once you apply it you obtain a new vector whose dot product is positive (this is by definition) but then you can relate this to angles.
however I think you can derive a condition for my problem by applying the Cramer Rule to the system.
The dot product is for the difference between any two vectors in the space and their difference in their corresponding transformations. This is slightly analogous to the first derivative. Let me see how the Cramer Rule may apply.
yes, it is a subdifferential. However given the nature of my problem I think you might want to consider the finite difference simply (which is essentially what I'm doing with Cramer really)
So you write $Ax = b$ you write $x$ as function of $b$ using the Cramer rule, you pick the difference between $x_{i+1}$ and $x_i$
but I can't see how to use my hypothesis explicitly
I mean given what you read (variational analysis) I guess you probably know better than I do that the subdifferential is somewhat meaningful in when the operator is not linear.
Discussion between user8469759 and prash
Discussion between user8469759 and prash