Hurwitz determinant - Alchetron, The Free Social Encyclopedia

In mathematics, Hurwitz determinants were introduced by Adolf Hurwitz (1895), who used them to give a criterion for all roots of a polynomial to have negative real part.

Definition

Let us consider a characteristic polynomial P in the variable λ of the form:

P ( λ ) = a 0 λ n + a 1 λ n − 1 + ⋯ + a n − 1 λ + a n

where a i , i = 0 , 1 , … , n , are real.

The square Hurwitz matrix associated to P is given below:

H = ( a 1 a 3 a 5 … … … 0 0 0 a 0 a 2 a 4 ⋮ ⋮ ⋮ 0 a 1 a 3 ⋮ ⋮ ⋮ ⋮ a 0 a 2 ⋱ 0 ⋮ ⋮ ⋮ 0 a 1 ⋱ a n ⋮ ⋮ ⋮ ⋮ a 0 ⋱ a n − 1 0 ⋮ ⋮ ⋮ 0 a n − 2 a n ⋮ ⋮ ⋮ ⋮ a n − 3 a n − 1 0 0 0 0 … … … a n − 4 a n − 2 a n ) .

The ith Hurwitz determinant is the determinant of the ith leading principal minor of the above Hurwitz matrix H. There are n Hurwitz determinants for a characteristic polynomial of degree n.

References

Hurwitz determinant Wikipedia

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