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.