Understanding Eigenvectors and Eigenvalues in Linear Algebra

[TheSpaceGuy]

I am trying to get an eigenvector for the following matrix, I am up to the final step.
4 1
0 0

I got it to be
-1
4

is this the same as
1
-4



sorry I am pretty new to linear algebra.

 

[Dick]

I am trying to get an eigenvector for the following matrix, I am up to the final step.
4 1
0 0

I got it to be
-1
4

is this the same as
1
-4
sorry I am pretty new to linear algebra.

Of course, they aren't the same vector. But if x is an eigenvector then c*x is also an eigenvector for any constant c. In your example the c is (-1). Both of those are fine eigenvectors.

 

[HallsofIvy]

As Dick said, any scalar multiple of an eigenvector is an eigenvector- in fact, any linear combination of eigenvectors is an eigenvector.

Of course, a good way to check if any vector is any eigenvector is to use the definition of "eigenvector". If v is an eigenvector of A, corresponding to eigenvalue $\lambda$, then $Av= \lambda v$.

Here,
$$\begin{bmatrix}4 & 1 \\ 0 & 0\end{bmatrix}\begin{bmatrix}-1 \\ 4\end{bmatrix}= \begin{bmatrix}-4+ 4 \\ 0\end{bmatrix}= \begin{bmatrix}0 \\ 0\end{bmatrix}= 0\begin{bmatrix}-1 \\ 4\end{bmatrix}$$
and
$$\begin{bmatrix}4 & 1 \\ 0 & 0\end{bmatrix}\begin{bmatrix} 1 \\ -4\end{bmatrix}= \begin{bmatrix}4- 4 \\ \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}= 0\begin{bmatrix}1 \\ -4\end{bmatrix}$$

So these are both eigenvectors corresponding to eigenvalue 0.

 

[TheSpaceGuy]

Thanks to the both of you. That really clears things up for me!