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A square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540801.png" /> with real entries such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540802.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540803.png" />. If one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540804.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540805.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540806.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540807.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540808.png" />), then a Jacobi matrix has the form
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{{MSC|47B36}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j0540809.png" /></td> </tr></table>
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A Jacobi matrix is a square matrix $[a_{i,k}]$ with real entries such that $a_{i,k} = 0$ for $\left|i-k\right|>1$. If one writes $a_{i,i} = a_i$ ($i=1,\ldots,n$), $a_{i,i+1}=b_i$, and $a_{i+1,i}=c_i$ ($i=1,\ldots,n-1$), then a Jacobi matrix has the form
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\[
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\left[
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  \begin{array}{cccccc}
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    a_1 & b_1 & 0   & \cdots & 0  & 0  \\
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    c_1 & a_2 & b_2 & \cdots & 0  & 0  \\
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    0  & c_2 & a_3 & \cdots & 0  & 0  \\
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    \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
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    0  & 0  & 0  & \cdots & a_{n-1} & b_{n-1} \\
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    0  & 0  & 0  & \cdots & c_{n-1} & a_n
  
Any [[Minor|minor]] of a Jacobi matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408010.png" /> is the product of certain principal minors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408011.png" /> and certain elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408012.png" />. A Jacobi matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408013.png" /> is completely non-negative (that is, all its minors are non-negative) if and only if all its principal minors and all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408016.png" />) are non-negative. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408018.png" />, then the roots of the [[Characteristic polynomial|characteristic polynomial]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054080/j05408019.png" /> are real and distinct.
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  \end{array}
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\right]
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\]
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Any [[Minor|minor]] of a Jacobi matrix $J$ is the product of certain principal minors of $J$ and certain elements of $J$. A Jacobi matrix $J$ is completely non-negative (that is, all its minors are non-negative) if and only if all its principal minors and all elements $b_i$ and $c_i$ ($i=1,\ldots,n-1$) are non-negative. If $b_ic_i>0$ for $i=1,\ldots,n-1$, then the roots of the [[Characteristic polynomial|characteristic polynomial]] of $J$ are real and distinct.
  
====References====
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====References====  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"F.R. Gantmakher,   M.G. Krein,   "Oscillation matrices and kernels and small vibrations of mechanical systems" , Dept. Commerce USA. Joint Publ. Service (1961) (Translated from Russian)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|GaKr}}||valign="top"| F.R. Gantmakher, M.G. Krein, "Oscillation matrices and kernels and small vibrations of mechanical systems", Dept. Commerce USA. Joint Publ. Service (1961) (Translated from Russian)
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|-
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|}

Revision as of 23:08, 23 July 2012

2010 Mathematics Subject Classification: Primary: 47B36 [MSN][ZBL]

A Jacobi matrix is a square matrix $[a_{i,k}]$ with real entries such that $a_{i,k} = 0$ for $\left|i-k\right|>1$. If one writes $a_{i,i} = a_i$ ($i=1,\ldots,n$), $a_{i,i+1}=b_i$, and $a_{i+1,i}=c_i$ ($i=1,\ldots,n-1$), then a Jacobi matrix has the form \[ \left[ \begin{array}{cccccc} a_1 & b_1 & 0 & \cdots & 0 & 0 \\ c_1 & a_2 & b_2 & \cdots & 0 & 0 \\ 0 & c_2 & a_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a_{n-1} & b_{n-1} \\ 0 & 0 & 0 & \cdots & c_{n-1} & a_n \end{array} \right] \] Any minor of a Jacobi matrix $J$ is the product of certain principal minors of $J$ and certain elements of $J$. A Jacobi matrix $J$ is completely non-negative (that is, all its minors are non-negative) if and only if all its principal minors and all elements $b_i$ and $c_i$ ($i=1,\ldots,n-1$) are non-negative. If $b_ic_i>0$ for $i=1,\ldots,n-1$, then the roots of the characteristic polynomial of $J$ are real and distinct.

References

[GaKr] F.R. Gantmakher, M.G. Krein, "Oscillation matrices and kernels and small vibrations of mechanical systems", Dept. Commerce USA. Joint Publ. Service (1961) (Translated from Russian)
How to Cite This Entry:
Jacobi matrix. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Jacobi_matrix&oldid=27190
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article