Suppose I have an n x n 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 ]

i.e., a companion matrix 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.

i.e., a companion matrix 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.