Eigenvectors

Given an eigenvalue $ \lambda$, nontrivial solutions of the homogeneous equation are called eigenvectors corresponding to $ \lambda$. The null space of  $ (A - \lambda I)$ is called the eigenspace corresponding to $ \lambda$.

Let $ \lambda_1,\ldots,\lambda_n$ be scalars. Then the $ n\times n$ square matrix

$\displaystyle D =
\begin{bmatrix}
\lambda_1 & 0 & \cdots & 0 \\
0 & \lambda_2 & \cdots & 0 \\
\hdotsfor{4} \\
0 & 0 & \cdots & \lambda_n
\end{bmatrix}$

is called the diagonal matrix with diagonal entries  $ \lambda_1,\ldots,\lambda_n$.

In Matlab/Octave, the function diag(a) with column vector a returns the diagonal matrix $ D$ with diagonal entries given by the vector a. The command

[V,D] = eig(A)
returns a diagonal matrix $ D$ with eigenvalues on diagonal entries, and a matrix $ V$ whose column vectors are the corresponding eigenvectors.

EXAMPLE 4. Find a basis for the eigenspace corresponding to the value $ \lambda$ in each of the following.

  1. $ A = \begin{bmatrix}
1 & 2 & 2 \\
3 &-2 & 1 \\
0 & 1 & 1
\end{bmatrix}$ with $ \lambda = 3$.
  2. $ A = \begin{bmatrix}
1 & 0 &-1 \\
1 &-3 & 0 \\
4 &-13& 1
\end{bmatrix}$ with $ \lambda = -2$.
  3. $ A = \begin{bmatrix}
3 & 0 & 2 & 0 \\
1 & 3 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 4 \\
\end{bmatrix}$ with $ \lambda = 4$.

EXAMPLE 5. Find the eigenvalues and the corresponding eigenvectors for each of the following matrices.

  1. $ A = \begin{bmatrix}
1 & 6 \\
5 & 2
\end{bmatrix}$
  2. $ A = \begin{bmatrix}
4 &-1 & 6 \\
2 & 1 & 6 \\
2 &-1 & 8
\end{bmatrix}$



Department of Mathematics
Last modified: 2005-10-21