HURWITZ MATRIX

In mathematics, a square matrix $A$ is called a 'Hurwitz matrix' if all eigenvalues of $A$ have strictly negative real part, that is,
:
for each eigenvalue $lambda\_i$. $A$ is also called a 'stability matrix', because then the differential equation
:$dot\; x\; =\; A\; x$
is stable, that is, $x(t)\; o\; 0$ as $t\; oinfty.$
If $G(s)$ is a (matrix-valued) transfer function, then $G$ is called 'Hurwitz' if the poles of all elements of $G$ have negative real part. Note that it is not necessary that $G(s),$ for a specific argument $s,$ be a Hurwitz matrix — it need not even be square. The connection is that if $A$ is a Hurwitz matrix, then the dynamical system
:$dot\; x(t)=A\; x(t)\; +\; B\; u(t)$
:$y(t)=C\; x(t)\; +\; D\; u(t),$
has a Hurwitz transfer function.

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References

★ Hassan K. Khalil (2002). ''Nonlinear Systems''. Prentice Hall.

★ Siegfried H. Lehnigk, ''On the Hurwitz matrix'', ''Zeitschrift für Angewandte Mathematik und Physik (ZAMP)'', May 1970

★ ''Hurwitz-Radon matrices revisited: From effective solution of the Hurwitz matrix equations to Bott periodicity'', in ''Mathematical Survey Lectures 1943–2004'', Springer Berlin Heidelberg, 2006

★ Bernard A. Asner, Jr., ''On the Total Nonnegativity of the Hurwitz Matrix'', SIAM Journal on Applied Mathematics, Vol. 18, No. 2 (Mar., 1970)

★ Dimitar K. Dimitrov and Juan Manuel Peña, ''Almost strict total positivity and a class of Hurwitz polynomials'', Journal of Approximation Theory, Volume 132, Issue 2 (February 2005)

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