A is a nonnegative matrix; the only principal submatrix having spectral radius above 1 is A itself

Let $\rho(M)$ denote the spectral radius (modulus of the largest eigenvalue) of a square matrix $M$.

I am looking for a characterization or anything else interesting about the set of matrices $A$ having the following properties:

(i) $A$ is (entrywise) nonnegative;

(ii) $A$ is irreducible;

(iii) $\rho(A) \geq 1$;

(iv) if $B$ is obtained from $A$ by deleting row $i$ and column $i$ (for any $i$), then $\rho(B) < 1$.

• What if diagonal entries are $\frac35$ and off diagonal $\frac15$? – Aaron Meyerowitz Oct 23 '12 at 17:06
• Shoot - too slow. I was gonna go with diagonal 2/3, off diagonal 1/4. – Zack Wolske Oct 23 '12 at 17:07
• If the matrix is irreducible with largest eigenvalue exactly $1$ then the property should hold. An easy way to achieve this is to have the entries on each row add to $1$. – Aaron Meyerowitz Oct 23 '12 at 17:14
• Deleted my last comment to fix an error: For 2x2 matrices, if you replace "trace > 1" by "trace > 1 + determinant", then it is a characterization. One still needs the diagonal entries to be less than 1, but there is no upper bound on the off-diagonals. For example, diagonal entries 3/4 and off-diagonal entries 2 satisfies (i)-(iv). – Zack Wolske Oct 23 '12 at 17:40

Let's assume the matrix $A$ has all its row sums equal to $\lambda$, the largest eigenvalue. We can rescale the rows and columns of any other nonnegative irreducible matrix by a similarity transformation $A \to D^{-1} A D$ where $D$ is diagonal with positive diagonal entries (namely the entries of the Perron eigenvector of $A$). This makes all row sums equal to $\lambda$, and preserves the eigenvalues of $A$ and of the matrices obtained by removing the $i$'th row and column.
Let $B_k$ be the matrix obtained from $A$ by removing row and column number $k$. The row sums of $B_k$ are $\lambda - a_{ik}$ for $i \ne k$. Thus the largest eigenvalue of $B_k$ is at most $\lambda - \min_{i \ne k} a_{ik}$ and at least $\lambda - \max_{i \ne k} a_{ik}$. So a necessary condition is $$\lambda \ge 1 > \lambda - \min_k \max_{i \ne k} a_{ik}$$ while a sufficient condition is $$\lambda \ge 1 > \lambda - \min_k \min_{i \ne k} a_{ik}$$
It is well-known that the study of irreducible nonnegative matrices is equivalent to studying (row) stochastic matrices (e.g., the paper by Johnson [Row stochastic matrices similar to doubly stochastic matrices. Linear and Multilinear Algebra 10 (1981), no. 2, 113–130; MR0618581]): 