Markov matrices are defined by non-negative entries where column sums equal unity, guaranteeing that one is always an eigenvalue and all other eigenvalues have a magnitude less than or equal to one; this spectral property ensures the system converges to a unique steady-state vector represented as the positive eigenvector corresponding to $\lambda=1$. Fourier series utilize an analogous principle within infinite-dimensional function spaces by expanding periodic functions into a basis of orthogonal trigonometric components (sines and cosines), where coefficients are determined via inner products defined as definite integrals over one period. Both concepts rely fundamentally on linear algebraic structures involving eigenspaces, null spaces, and orthonormal bases to describe dynamical equilibrium states in discrete stochastic systems and continuous function approximations respectively.
A Markov matrix has nonnegative entries with each column summing to one, a structure that guarantees λ = 1 is always an eigenvalue (since the columns of A − I sum to zero, making it singular) and tha…