# Linear system with only positive solutions

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• Mar 24th 2011, 03:45 AM
pesius
Linear system with only positive solutions
Hi,
I'm involved in the dimensioning of a markov chain.
In order to find its state probabilities I have constructed a system of equation, resulting in a sparse matrix of coefficients of the type Ax = b.

The problem is that when I solve the system (the solutions are correct, I already checked it), some states have negative solutions (which means that the probability of being in these states are negative (Headbang)).

Is there any rule/condition/property that the matrix A must have in order to obtain only Zero or Positive solutions?

Thanks in advance,
G.B.
• Mar 24th 2011, 11:05 AM
dwsmith
Quote:

Originally Posted by pesius
Hi,
I'm involved in the dimensioning of a markov chain.
In order to find its state probabilities I have constructed a system of equation, resulting in a sparse matrix of coefficients of the type Ax = b.

The problem is that when I solve the system (the solutions are correct, I already checked it), some states have negative solutions (which means that the probability of being in these states are negative (Headbang)).

Is there any rule/condition/property that the matrix A must have in order to obtain only Zero or Positive solutions?

Thanks in advance,
G.B.

Can you post what you have done? The probabilities of the columns of a Markov chain matrix are nonnegative and add up to 1.
• Mar 28th 2011, 12:58 AM
pesius
Quote:

Originally Posted by dwsmith
Can you post what you have done? The probabilities of the columns of a Markov chain matrix are nonnegative and add up to 1.

The probabilities add up to 1, but some of them are negative.

Reading again my initial post I've seen that the problem is not well explained.

The linear system is of the type Ax = b, where A is a sparse matrix, b is a column vector of all zeros (only the last element is 1) and x is a column vector that contains the states probabilities.

The vector x that must contains only positive solutions and I would like to find out some properties that A should respects to guarantee that condition.