when will all eigenvalues be less then 1?

Are there any sufficient conditions to guarantee that all eigenvalues of an $n$ by $n$ matrix are less than $1$?

I tried to use "the sum of each row is less than 1", but it's still complicated for my matrix. The matrix I have is actually a Jacobian matrix, I'd like to prove that a dynamic system is locally stable in the steady states, thus I may not able to calculate the eigenvalues analytically.

Solution 1:

Well, sum of absolute values of each row being less than $1$ does work. Look up http://mathworld.wolfram.com/GershgorinCircleTheorem.html .

Also, if your matrix is real and positive, you could try to see if the Perron Frobenius theorem applies. This way, if all the real, positive, simple eigenvalues are smaller than $1$, then all eigenvalues have absolute values less than 1.

Can you divulge the kind of matrices you want to apply such a criterion to? It doesn't seem that one universal criterion is applicable in all cases.