What is recorded here?
For a real polynomial with positive leading coefficient, positivity of specified coefficient determinants characterizes whether every root has strictly negative real part. A linear model can grow or decay depending on where its characteristic roots lie. Finding every root can be inconvenient when coefficients contain design parameters. Hurwitz replaces that calculation with a finite list of determinant signs, obtaining a necessary and sufficient test for the stated polynomial problem. The engineering question came from Stodola; the proof develops inherited Cauchy-index and quadratic-form ideas. Earlier stability work, including Routh’s line, prevents treating this as the first conceivable test. Applying the criterion to a physical controller still requires a faithful model; passing a polynomial test alone is not proof that a nonlinear or uncertain machine will be safe.
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Scientific scope
Real-coefficient degree-n polynomial with positive leading coefficient: all roots have strictly negative real part iff the specified Hurwitz determinants are positive. All positive coefficients are necessary, not sufficient by themselves. The general Cauchy-index count first excludes imaginary-axis roots; the final special half-count convention is not a strict-stability guarantee for boundary roots. Finite-degree coefficient mathematics does not certify a nonlinear, uncertain, delayed or discretized plant without its own model/conditions. Original application footnote is author-reported, not independently replayed engineering success. matrix entries, signs, coefficients and full proof not independently certified. Earlier Maxwell/Routh and later equivalent forms are not absorbed as first discoveries; manuscript 1894 differs from publication 1895.
Read the original work
- Hurwitz, negative-real-part root criterion (1895), original273–284
- ETH Library institutional archive portrait of Adolf Hurwitz
- ETH Automatic Control Laboratory: Stodola/Hurwitz historical paragraph
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People and systems
Adolf Hurwitz
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English catalogue narrative
Hurwitz: checking decay without solving every polynomial root
For a real polynomial with positive leading coefficient, positivity of specified coefficient determinants characterizes whether every root has strictly negative real part.
A linear model can grow or decay depending on where its characteristic roots lie. Finding every root can be inconvenient when coefficients contain design parameters. Hurwitz replaces that calculation with a finite list of determinant signs, obtaining a necessary and sufficient test for the stated polynomial problem. The engineering question came from Stodola; the proof develops inherited Cauchy-index and quadratic-form ideas. Earlier stability work, including Routh’s line, prevents treating this as the first conceivable test. Applying the criterion to a physical controller still requires a faithful model; passing a polynomial test alone is not proof that a nonlinear or uncertain machine will be safe.
Scientific scope
Real-coefficient degree-n polynomial with positive leading coefficient: all roots have strictly negative real part iff the specified Hurwitz determinants are positive. All positive coefficients are necessary, not sufficient by themselves. The general Cauchy-index count first excludes imaginary-axis roots; the final special half-count convention is not a strict-stability guarantee for boundary roots. Finite-degree coefficient mathematics does not certify a nonlinear, uncertain, delayed or discretized plant without its own model/conditions. Original application footnote is author-reported, not independently replayed engineering success. matrix entries, signs, coefficients and full proof not independently certified. Earlier Maxwell/Routh and later equivalent forms are not absorbed as first discoveries; manuscript 1894 differs from publication 1895.
People and systems
Adolf Hurwitz