Definition:Homogeneous Linear Equations

Definition

A system of homogeneous linear equations is a set of simultaneous linear equations:

$\displaystyle \forall i \in \left[{1 .. m}\right] : \sum_{j=1}^n \alpha_{i j} x_j = \beta_i$

such that all the $\beta_i$ are equal to zero:

$\displaystyle \forall i \in \left[{1 .. m}\right] : \sum_{j=1}^n \alpha_{i j} x_j = 0$

That is:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 0\)	\(=\)	\(\displaystyle \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 0\)	\(=\)	\(\displaystyle \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\cdots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 0\)	\(=\)	\(\displaystyle \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Matrix Representation

Such a system is often expressed as:

$ \mathbf A \mathbf x = \mathbf 0$

where:

$ \mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$, $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf 0 = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$

are matrices.

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.4$