> "It turns out that a matrix like A, whose rows add up to zero (try it!), is called a Markov matrix, ..."
Oops, you mean the rows add to one.
I hate to nitpick, but, additionally, numbers in the matrix can't be negative.
Also, it's not just that 1 is an eigenvalue, it is that 1 is the largest eigenvalue. This is significant, because it implies that all other components will die out in time.
The cited source says that in a Markov matrix the columns sum to 1 which Wikipedia reports is a "left stochastic matrix" whilst the rows summing to 1 makes it a "right stochastic matrix" - I'm not sure on the definition of Markov Matrix specifically but then it doesn't seem that http://mathworld.wolfram.com/StochasticMatrix.html knows either. The term returns no search results at http://www.encyclopediaofmath.com/.
Whether it's rows vs. columns is just a matter of definition, depending on whether you want to propagate the state by left multiplication or right multiplication. The typical definition (in the context of Markov chains) has the rows summing to one. But how to do it is up to the author of the page. Just be consistent.
In my own coursework, that matrix is called a "stochastic matrix", btw, not a "Markov matrix", but again, it's just definitional and not of interest in a simple article like this.
My main point is that summing to one is what you want, and summing to zero is crazy.
Comments
Oops, you mean the rows add to one.
I hate to nitpick, but, additionally, numbers in the matrix can't be negative.
Also, it's not just that 1 is an eigenvalue, it is that 1 is the largest eigenvalue. This is significant, because it implies that all other components will die out in time.
The cited source says that in a Markov matrix the columns sum to 1 which Wikipedia reports is a "left stochastic matrix" whilst the rows summing to 1 makes it a "right stochastic matrix" - I'm not sure on the definition of Markov Matrix specifically but then it doesn't seem that http://mathworld.wolfram.com/StochasticMatrix.html knows either. The term returns no search results at http://www.encyclopediaofmath.com/.
http://blog.stata.com/2011/03/09/understanding-matrices-intu... came up here before and helped me visualise eigen-stuff.
Whether it's rows vs. columns is just a matter of definition, depending on whether you want to propagate the state by left multiplication or right multiplication. The typical definition (in the context of Markov chains) has the rows summing to one. But how to do it is up to the author of the page. Just be consistent.
In my own coursework, that matrix is called a "stochastic matrix", btw, not a "Markov matrix", but again, it's just definitional and not of interest in a simple article like this.
My main point is that summing to one is what you want, and summing to zero is crazy.
Indeed, I've heard of stochastic matrices - just wasn't sure they were coterminous in definition with "Markov matrix". Thanks for clarifying though.