Two distinct Eigenvectors corresponding to the same Eigenvalue are always linearly dependent.
So i know this statement is false, but i don't understand the reason given by the book.
"" The vectors (1,0), (2,0) are both the eigenvectors of the matrix
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
corresponding to the same eignevalue 1.""
Aren't both vectors are linearly dependent?
Best Answer
Well, if that's the example, change book! :D Jokes aside, those two vectors are indeed linearly dependent. For an example of independent eigenvectors, you can instead consider the matrix $$\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}$$ and note that the vectors $$\begin{pmatrix}1\\-1\\0\end{pmatrix}\;,\ \ \begin{pmatrix}1\\0\\-1\end{pmatrix}$$ are both eigenvectors for the eigenvalue $-1$. In general, the eigenvectors for the eigenvalue $\lambda$ are the elements of $\ker(A-\lambda I)$; if $\dim\ker(A-\lambda I)>1$, then by definition you can find at least two independent vectors inside it.