Stability of a Companion Matrix

Linearizing a nonlinear difference
equation within neighborhood of its fixed
point yields an n x n companion matrix A =

[ 0 1 0 ... 0 ]
[ 0 0 1 ... 0 ]
[ 0 0 0 ... 0 ]
[ ... ... ... ... ... ]
[ 0 0 0 0 1 ]
[ x_1 x_2 x_3 ... x_n ]

where each of the elements of the last row is

x_i = [1 - 1 / (1 + r)] a_i
a_i = any real number

for i = 1 ... n.

If r = 0, then x_i = 0 for all i and
the matrix has all eigenvalues at the
origin.

My question is, what is the
range of r such that every eigenvalue
of the matrix is within the unit circle?

Thanks.


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