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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 $\rho(B) < 1$.

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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 $$\lambda \ge 1 > \lambda - \min_k \min_{i \ne k} a_{ik}$$

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Thanks -- this is very helpful. I especially like the elegant idea of working with an A that has all row sums equal. I wonder if it's possible to use this idea to think about similar conditions involving powers of A and B, where the conditions might be less demanding than the sufficient condition (i.e. they could bite for sparse matrices... – Ben Golub Oct 23 '12 at 22:41

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