Matrix Algebra

Homogeneous Equations

Homogeneous equations. We define the zero vector in $\mathbb{R}^m$ by $\left[\begin{array}{c} 0 \\ \vdots \\ 0 \\ \end{array}\right]$ (whose entries are all zero's), and denote it by $\mathbf{0}$ . Let

be an $m\times n$ matrix, Then we can define a homogeneous equation $A\mathbf{x} = \mathbf{0}$ . Note that it is a special case of the matrix equation $A \mathbf{x} = \mathbf{b}$ with $\mathbf{b} = \mathbf{0}$ in $\mathbb{R}^m$ . The homogeneous equation $A\mathbf{x} = \mathbf{0}$ has always the trivial solution $\mathbf{x} = \mathbf{0}$ in $\mathbb{R}^n$ , but may have nontrivial solutions $\mathbf{x} \neq \mathbf{0}$ .

EXAMPLE 1. Determine whether the homogeneous system

$\begin{displaymath} \left\{ \begin{array}{rrrr} 3x_1 & +5x_2 & -4x_3 = & 0 \... ... = & 0 \\ 6x_1 & + x_2 & -8x_3 = & 0 \end{array} \right. \end{displaymath}$

has nontrivial solutions. Then describe the solution set.

Matlab/Octave. The function zeros(m,1) produces the -dimensional zero column vector.