System of Linear Equations - Homogeneous Systems

Homogeneous Systems

See also: Homogeneous differential equation

A system of linear equations is homogeneous if all of the constant terms are zero:

\begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = \;&&& 0 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = \;&&& 0 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \,\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = \;&&& 0. \\ \end{alignat}

A homogeneous system is equivalent to a matrix equation of the form

where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.

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