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| In mathematics, '''Hurwitz determinants''' were introduced by {{harvs|txt|authorlink=Adolf Hurwitz|first=Adolf|last= Hurwitz|year=1895}}, who used them to give a criterion for all roots of a polynomial to have negative real part.
| | My name's Mallory Webb but everybody calls me Mallory. I'm from Austria. I'm studying at the college (3rd year) and I play the Piano for 3 years. Usually I choose songs from the famous films ;). <br>I have two sister. I love Games Club - Dungeons and Dragons, Monopoly, Etc., watching movies and Homebrewing.<br><br>Feel free to visit my web page; [http://www.netcrew.co.za/index.php?mod=users&action=view&id=111575 can someone write my paper for me] |
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| ==Definition==
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| Let us consider a [[characteristic polynomial]] ''P'' in the variable ''λ'' of the form:
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| :<math>
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| P(\lambda)= a_0 \lambda^n + a_1 \lambda^{n-1} + \cdots + a_{n-1} \lambda + a_n
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| </math>
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| where <math>a_i</math>, <math>i=0,1,\ldots,n</math>, are real.
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| The square [[Hurwitz matrix]] associated to ''P'' is given below:
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| :<math>
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| H=
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| \begin{pmatrix}
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| a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\
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| a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\
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| 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\
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| \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\
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| \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\
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| \vdots & \vdots & a_0 & & & \ddots & a_{n-1} & 0 & \vdots \\
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| \vdots & \vdots & 0 & & & & a_{n-2} & a_n & \vdots \\
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| \vdots & \vdots & \vdots & & & & a_{n-3} & a_{n-1} & 0 \\
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| 0 & 0 & 0 & \dots & \dots & \dots & a_{n-4} & a_{n-2} & a_n
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| \end{pmatrix}.
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| </math> | |
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| The ''i''th ''Hurwitz determinant'' is the determinant of the ''i''th [[principal minor]] of the above Hurwitz matrix ''H''. There are ''n'' Hurwitz determinants for a characteristic polynomial of degree ''n''.
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| ==See also==
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| * [[Transfer matrix]]
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| ==References==
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| *{{Citation | last1=Hurwitz | first1=A. | title=Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt | doi= 10.1007/BF01446812 | year=1895 | journal= Mathematische Annalen | volume=46 | issue=2}}
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| *{{Citation | last1=Wall | first1=H. S. | title=Polynomials whose zeros have negative real parts | jstor=2305291 | mr=0012709 | year=1945 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=52 | pages=308–322}}
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| [[Category:Linear algebra]]
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| [[de:Hurwitzpolynom#Hurwitz-Kriterium]]
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My name's Mallory Webb but everybody calls me Mallory. I'm from Austria. I'm studying at the college (3rd year) and I play the Piano for 3 years. Usually I choose songs from the famous films ;).
I have two sister. I love Games Club - Dungeons and Dragons, Monopoly, Etc., watching movies and Homebrewing.
Feel free to visit my web page; can someone write my paper for me