Let's say that we have a matrix $A$ that is a square matrix and has rows that sum up to 1 (think of a transition matrix in Markov chains). In this case, the largest absolute eigenvalue will be 1, while the rest of the eigenvalues will be less than one (Proof: Proof that the largest eigenvalue of a stochastic matrix is 1).
Is there a similar constraint for non-square matrices that would force the largest absolute singular value to be one, while forcing the other singular values to be all less than 1 in magnitude?
Thank you.
