The Eigenvectors(A) command returns an expression sequence of two elements. The first element is the Vector of eigenvalues (that is, exactly what is returned by the Eigenvalues(A) command.) The second is the Matrix of corresponding eigenvectors.

For example, after

(ev, EV) := Eigenvectors(A);

for each column index i, $A·{\mathrm{EV}}_{1..\mathrm{-1},i}={\mathrm{ev}}_i·{\mathrm{EV}}_{1..\mathrm{-1},i}$.

Note: If the input Matrix A is defective (does not have a full set of linearly independent eigenvectors) then some of the columns of the Matrix of eigenvectors are 0 (and hence are not eigenvectors).  See JordanForm.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$A≔⟨⟨-1\,-3\,-6⟩\left|⟨3\,5\,6⟩\right|⟨-3\,-3\,-4⟩⟩$

$A≔\left[\begin{array}{ccc}\mathrm{-1}& 3& \mathrm{-3}\\ \mathrm{-3}& 5& \mathrm{-3}\\ \mathrm{-6}& 6& \mathrm{-4}\end{array}\right]$

$\mathrm{ev},\mathrm{EV}≔\mathrm{Eigenvectors}\left(A\right)$

$\mathrm{ev},\mathrm{EV}≔\left[\begin{array}{c}2\\ 2\\ \mathrm{-4}\end{array}\right],\left[\begin{array}{ccc}\mathrm{-1}& 1& \frac{1}{2}\\ 0& 1& \frac{1}{2}\\ 1& 0& 1\end{array}\right]$

$A·\mathrm{EV}\left[1..-1,2\right]=\mathrm{ev}\left[2\right]·\mathrm{EV}\left[1..-1,2\right]$

$\left[\begin{array}{c}2\\ 2\\ 0\end{array}\right]=\left[\begin{array}{c}2\\ 2\\ 0\end{array}\right]$