This is close to some of my favorite stuff in math!
Beyond just markov chains, matrices do “move” vectors towards eigenvectors sometimes. If a matrix A has eigenvectors x1 and x2 with eigenvalues r1 and r2, A(x1+x2)= r1x1+r2x2 because matrix multiplication is a linear transformation. If we repeatedly multiply x1+x2 by A, A^n(x1+x2)=r1^nx1+ r2^nx2. Then, if r1>r2, all of the terms are growing exponentially but the contribution of x1 to the result grows exponentially faster than the contribution of x2 so for some large n, A^n(x1+x2) = r1^nx1 + some irrelevant error term.
This means the largest eigenvalues sort of dominate, and you might especially care about eigenvalues of 1, because those mean Ax=x so x is a steady state and if you can write down A as a matrix you can solve for non zero x and learn about the steady state solutions.
Beyond just markov chains, matrices do “move” vectors towards eigenvectors sometimes. If a matrix A has eigenvectors x1 and x2 with eigenvalues r1 and r2, A(x1+x2)= r1x1+r2x2 because matrix multiplication is a linear transformation. If we repeatedly multiply x1+x2 by A, A^n(x1+x2)=r1^nx1+ r2^nx2. Then, if r1>r2, all of the terms are growing exponentially but the contribution of x1 to the result grows exponentially faster than the contribution of x2 so for some large n, A^n(x1+x2) = r1^nx1 + some irrelevant error term.
This means the largest eigenvalues sort of dominate, and you might especially care about eigenvalues of 1, because those mean Ax=x so x is a steady state and if you can write down A as a matrix you can solve for non zero x and learn about the steady state solutions.