The function eig(A) computes the eigenvalues and eigenvectors of the matrix A. That is, for each eigenvalue lambda of A, it solves the linear system (I * lambda - A) * X= 0  for X.

When the function is called using the form l := eig(A), the returned value of l is a column Vector containing the eigenvalues of A.

When the function is called using the form V,L := eig(A), the returned value of L is a Matrix with the eigenvalues of A along the main diagonal, and the returned value of V is a Matrix whose columns are the eigenvectors of A.

When the function is called using the form V,L,N := eig(A), L and V are as described above. N is a row vector of indices, one for each linearly independent eigenvector of A, such that the vector corresponding to the ith column of V has eigenvalue L[N[i],N[i]].

$\mathrm{with}\left(\mathrm{MTM}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\,1\right]\,\left[2\,4\,2\right]\,\left[2\,8\,1\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 2& 8& 1\end{array}\right]$

$l≔\mathrm{eig}\left(A\right)$

$l≔\left[\begin{array}{c}0\\ 3+\sqrt{22}\\ 3-\sqrt{22}\end{array}\right]$

$V,L≔\mathrm{eig}\left(A\right)$

$V,L≔\left[\begin{array}{ccc}\frac{9\left(3+\sqrt{22}\right)}{\left(-5+7\sqrt{22}\right)\left(2+\sqrt{22}\right)}& \frac{9\left(3-\sqrt{22}\right)}{\left(-5-7\sqrt{22}\right)\left(2-\sqrt{22}\right)}& -\frac{3}{2}\\ \frac{16+\sqrt{22}}{-5+7\sqrt{22}}& \frac{16-\sqrt{22}}{-5-7\sqrt{22}}& \frac{1}{4}\\ 1& 1& 1\end{array}\right],\left[\begin{array}{ccc}3+\sqrt{22}& 0& 0\\ 0& 3-\sqrt{22}& 0\\ 0& 0& 0\end{array}\right]$

$V,L,N≔\mathrm{eig}\left(A\right)$

$V,L,N≔\left[\begin{array}{ccc}\frac{9\left(3+\sqrt{22}\right)}{\left(-5+7\sqrt{22}\right)\left(2+\sqrt{22}\right)}& \frac{9\left(3-\sqrt{22}\right)}{\left(-5-7\sqrt{22}\right)\left(2-\sqrt{22}\right)}& -\frac{3}{2}\\ \frac{16+\sqrt{22}}{-5+7\sqrt{22}}& \frac{16-\sqrt{22}}{-5-7\sqrt{22}}& \frac{1}{4}\\ 1& 1& 1\end{array}\right],\left[\begin{array}{ccc}3+\sqrt{22}& 0& 0\\ 0& 3-\sqrt{22}& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$