Eigenvectors for the matrix

Question

Eigenvectors for $\begin{pmatrix}1&1\\ \:-5&-3\end{pmatrix}$

Eigenvalues are $-1+i$ and $-1-i$

• We calculate for $-1+i$

$\begin{pmatrix}1-\left(-1+i\right)&1\\ \:-5&-3-\left(-1+i\right)\end{pmatrix}=\begin{pmatrix}2-i&1\\ \:\:-5&-2-i\end{pmatrix}=\begin{pmatrix}4-2i&2\\ \:\:\:-5&-2-i\end{pmatrix}$

We add the first and the second row together

$\left(-1-2i\right)y_1-iy_2=0$

We get

$-iy_2=\left(1+2i\right)y_1\Leftrightarrow y_2=\left(i-2\right)y_1$

Making

$\begin{pmatrix}y\\ \left(i-2\right)y\end{pmatrix}$

The eigenvector for eigenvalue $-1+i$

Is this true?

Answer 1

Yes, you can always check the validity of your answer by multiplying the vector you ended up with, with the matrix and hence check if it satisfies the desired conditions: (the image vector being a constant (eigenvalue) multiple of your vector.