Difference between revisions of "Jacobi matrix"
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{{MSC|47B36}} | {{MSC|47B36}} | ||
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| + | ===Linear algebra=== | ||
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 | 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 | ||
\[ | \[ | ||
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\] | \] | ||
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. | 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}{llll} | ||
| + | \frac{\partial f^1}{\partial x_1} (y) & \frac{\partial f^1}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^1}{\partial x_n} (y)\\ | ||
| + | \frac{\partial f^2}{\partial x_1} (y) & \frac{\partial f^2}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^2}{\partial x_n} (y)\\ | ||
| + | \\ | ||
| + | \vdots & \vdots & &\vdots\\ | ||
| + | \\ | ||
| + | \frac{\partial f^m}{\partial x_1} (y) & \frac{\partial f^m}{\partial x_2} (y)&\qquad \ldots \qquad & \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==== | ====References==== | ||
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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) | |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}} | ||
|- | |- | ||
|} | |} | ||
Revision as of 09:12, 11 December 2012
2020 Mathematics Subject Classification: Primary: 47B36 [MSN][ZBL]
Linear algebra
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.
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}{llll} \frac{\partial f^1}{\partial x_1} (y) & \frac{\partial f^1}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^1}{\partial x_n} (y)\\ \frac{\partial f^2}{\partial x_1} (y) & \frac{\partial f^2}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^2}{\partial x_n} (y)\\ \\ \vdots & \vdots & &\vdots\\ \\ \frac{\partial f^m}{\partial x_1} (y) & \frac{\partial f^m}{\partial x_2} (y)&\qquad \ldots \qquad & \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 |
Jacobi matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_matrix&oldid=27190