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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
 
  
<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>
 
  
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.
+
{{MSC|47B36}}
 +
{{TEX|done}}
  
====References====
+
===Linear algebra===
<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>
+
A Jacobi matrix is a square [[tridiagonal 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|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.
 +
 
 +
===Analysis===
 +
Let $U\subset \mathbb R^n$, $f: U\to \mathbb R^m$ and assume that $f$ is differentiable at the point $y\in U$.
 +
The Jacobi matrix, or [[Jacobian]], of $f$ at $y$ is then the matrix
 +
\begin{equation}\label{e:Jacobi_matrix}
 +
Df|_y := \left(
 +
\begin{array}{cccc}
 +
\frac{\partial  f^1}{\partial x_1} (y) & \frac{\partial f^1}{\partial x_2}  (y) & \cdots  & \frac{\partial f^1}{\partial x_n}  (y)\\
 +
\frac{\partial f^2}{\partial x_1} (y) & \frac{\partial  f^2}{\partial  x_2} (y) & \cdots  & \frac{\partial  f^2}{\partial  x_n} (y)\\
 +
\vdots & \vdots & \ddots &\vdots\\
 +
\frac{\partial  f^m}{\partial x_1} (y) & \frac{\partial f^m}{\partial  x_2}  (y) & \cdots  & \frac{\partial f^m}{\partial  x_n}  (y)
 +
\end{array}\right)\, ,
 +
\end{equation}
 +
where $(f^1,  \ldots, f^m)$ are the coordinate functions of $f$ and $x_1,\ldots, x_n$  denote the standard system of coordinates in
 +
$\mathbb R^n$.
 +
 
 +
 
 +
====References====  
 +
{|
 +
|-
 +
|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)
 +
|-
 +
|valign="top"|{{Ref|Ru}}|| W. Rudin, "Principles of  mathematical  analysis", Third edition, McGraw-Hill (1976) {{MR|038502}}    {{ZBL|0346.2600}} 
 +
|-
 +
|}

Latest revision as of 01:37, 7 May 2022


2020 Mathematics Subject Classification: Primary: 47B36 [MSN][ZBL]

Linear algebra

A Jacobi matrix is a square tridiagonal 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.

Analysis

Let $U\subset \mathbb R^n$, $f: U\to \mathbb R^m$ and assume that $f$ is differentiable at the point $y\in U$. The Jacobi matrix, or Jacobian, of $f$ at $y$ is then the matrix \begin{equation}\label{e:Jacobi_matrix} Df|_y := \left( \begin{array}{cccc} \frac{\partial f^1}{\partial x_1} (y) & \frac{\partial f^1}{\partial x_2} (y) & \cdots & \frac{\partial f^1}{\partial x_n} (y)\\ \frac{\partial f^2}{\partial x_1} (y) & \frac{\partial f^2}{\partial x_2} (y) & \cdots & \frac{\partial f^2}{\partial x_n} (y)\\ \vdots & \vdots & \ddots &\vdots\\ \frac{\partial f^m}{\partial x_1} (y) & \frac{\partial f^m}{\partial x_2} (y) & \cdots & \frac{\partial f^m}{\partial x_n} (y) \end{array}\right)\, , \end{equation} where $(f^1, \ldots, f^m)$ are the coordinate functions of $f$ and $x_1,\ldots, x_n$ denote the standard system of coordinates in $\mathbb R^n$.


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)
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
How to Cite This Entry:
Jacobi matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_matrix&oldid=17361
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article