克罗内克函数

$\delta_{ij} = \left\{\begin{matrix} 1 & (i=j) \\ 0 & (i \ne j) \end{matrix}\right.\,\!$

其它记法

$\delta_{ij} = [i=j ]\,\!$

$\delta_{i} = \left\{\begin{matrix} 1, & \mbox{if } i=0 \\ 0, & \mbox{if } i \ne 0 \end{matrix}\right.\,\!$

数字信号处理

$\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0\end{cases}\,\!$

性质

$\sum_{i= - \infty}^\infty \delta_{ij} a_i=a_j\,\!$

$\int_{-\infty}^\infty \delta(x-y)f(x) dx=f(y)\,\!$

线性代数中的应用

• 作为线性映射的单位矩阵。
• 迹数
• 内积 $V^* \otimes V \to K\,\!$
• 映射 $K \to V^* \otimes V\,\!$ ，将数量乘积表示为外积的形式。

廣義克羅內克函數

$\delta^{j_1 j_2 \dots j_n}_{i_1 i_2 \dots i_n} = \begin{bmatrix} \delta^{j_1}_{i_1} \delta^{j_1}_{i_2} & \cdots & \delta^{j_1}_{i_n} \\ \delta^{j_2}_{i_1} \delta^{j_2}_{i_2} & \cdots & \delta^{j_2}_{i_n} \\ \vdots & \ddots & \vdots \\ \delta^{j_n}_{i_1} \delta^{j_n}_{i_2} & \cdots & \delta^{j_n}_{i_n} \\ \end{bmatrix}\,\!$

• $\delta^{ijk}_{imn} =\delta^{jk}_{mn}=\delta^{j}_{m}\delta^{k}_{n} - \delta^{j}_{n}\delta^{k}_{m}\,\!$
• $\delta^{ijk}_{ijm} =2\delta^{k}_{m}\,\!$
• $\delta^{ijk}_{ijk} =6\,\!$
• $\delta^{ijk}_{lmn} =\epsilon^{ijk}\epsilon_{lmn}\,\!$

• $\delta^{j_1 j_2 \dots j_n}_{i_1 i_2 \dots i_n} =\epsilon^{j_1 j_2 \dots j_n}\epsilon_{i_1 i_2 \dots i_n}\,\!$
• $\delta^{1 2 \dots n}_{i_1 i_2 \dots i_n} =\epsilon_{i_1 i_2 \dots i_n}\,\!$
• $\delta^{j_1 j_2 \dots j_n}_{i_1 i_2 \dots i_n} T_{j_1 j_2 \dots j_n}=n!\ T_{i_1 i_2 \dots i_n}\,\!$

积分表示

$\delta_{x,n} = \frac1{2\pi i} \oint z^{x-n-1} dz\,\!$

$\delta_{x,n} = \frac1{2\pi} \int_0^{2\pi} e^{i(x-n)\varphi} d\varphi\,\!$

參考文獻

1. ^ Heinbockel, J. H., Introduction to Tensor Calculus and Continum Mechanics, Victoria, B.C. Canada: Trafford Publishing, pp. 14, 31, 2001, ISBN 1-55369-1333-4 请检查|isbn=值 (帮助)