The characteristic roots of a matrix

If A is a square matrix of order n and I is the unit matrix, the equation in X obtained by equating to zero the determinant \A— \l\ is called the characteristic equation of A. The roots of this equation are called the characteristic roots of A. Although it is not possible to make any definite statement as to the nature of the characteristic roots of the general algebraic matrix A, several authors have given upper limits to the roots. The first upper limit seems to have been given by Bendixsonf in 1900. Let us denote by A f and A the transpose and the conjugate imaginary, respectively, of the square matrix A. If we write