Hurwitz determinant

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

Consider a characteristic polynomial P in the variable λ of the form:

P(λ)=a0λn+a1λn−1+⋯+an−1λ+an

where ai, i=0,1,…,n, are real.

The square Hurwitz matrix associated to P is given below:

H=(a1a3a5………000a0a2a4⋮⋮⋮0a1a3⋮⋮⋮⋮a0a2⋱0⋮⋮⋮0a1⋱an⋮⋮⋮⋮a0⋱an−10⋮⋮⋮0an−2an⋮⋮⋮⋮an−3an−10000………an−4an−2an).

The i-th Hurwitz determinant is the i-th leading principal minor (minor is a determinant) of the above Hurwitz matrix H. There are n Hurwitz determinants for a characteristic polynomial of degree n.

## See also

* Transfer matrix

## References

* Hurwitz, A. (1895), "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt", Mathematische Annalen 46 (2): 273–284, doi:10.1007/BF01446812
* Wall, H. S. (1945), "Polynomials whose zeros have negative real parts", The American Mathematical Monthly 52 (6): 308–322, doi:10.1080/00029890.1945.11991574, ISSN 0002-9890

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