Eigenvectors and eigenvalues

Let $ A$ be an $ m\times n$ matrix, and let $ \mathbf{0}$ be the zero vector in  $ \mathbb{R}^m$ whose entries are all zero's. Then we can define a homogeneous equation  $ A\mathbf{x} = \mathbf{0}$, which is a special case of matrix equation $ A \mathbf{x} = \mathbf{b}$ with $ \mathbf{b} = \mathbf{0}$. The homogeneous equation  $ A\mathbf{x} = \mathbf{0}$ has always the trivial solution  $ \mathbf{x} = \mathbf{0}$, but may have nontrivial solutions  $ \mathbf{x} \neq \mathbf{0}$.

EXAMPLE 1. Determine whether the homogeneous system

\begin{displaymath}
\begin{array}{rrrr}
3x_1 & +5x_2 & -4x_3 = & 0 \\
-3x_1 & -2x_2 & +4x_3 = & 0 \\
6x_1 & + x_2 & -8x_3 = & 0
\end{array}\end{displaymath}

has a nontrivial solution. Then describe the solution set.

Let $ A$ be an $ n\times n$ square matrix, and let $ I$ be the $ n\times n$ identity matrix. Let $ \lambda$ be a scalar. If the homogeneous equation

$\displaystyle (A - \lambda I)\mathbf{x} = \mathbf{0}$ (9)

has nontrivial solutions, then the scalar $ \lambda$ is called an eigenvalue of $ A$.

In Matlab/Octave the function eig(A) returns a vector containing eigenvalues of $ A$.



Subsections

Department of Mathematics
Last modified: 2005-10-21