Jacobi matrix

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