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Jacobi matrix

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

Linear algebra

A Jacobi matrix is a square tridiagonal matrix 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=52310
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article