Distribution of zeros of derivatives of characteristic polynomials
The eigenvalues of a unitary (or orthogonal) matrix lie on the unit circle in the field of complex numbers. Consequently, the zeros of the derivative of the characteristic polynomial of the matrix lie strictly inside the unit circle (other than those coming from multiple zeros).
How are the zeros of the derivative distributed? What is the distribution of the absolute value of those zeros?
One does not generate a uniformly random collection of unitary of orthogonal matrices by choosing the entries uniformly randomly: see Orthogonal Matrix (section on Randomization) and Stewart (1980), below. Basically, one constructs a matrix with independent normally distributed random entries, then uses QR decomposition.
References & readings
- G. W. Stewart (1980) The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators. SIAM Journal on Numerical Analysis, Vol. 17(3), 403-409.