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 eig
 compute the eigenvalues and eigenvectors of a matrix

 Calling Sequence l = eig(A) [V,L] = eig(A) [V,L,N] = eig(A)

Parameters

 A - matrix

Description

 • 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]].

Examples

 > $\mathrm{with}\left(\mathrm{MTM}\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,2,1\right],\left[2,4,2\right],\left[2,8,1\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{rrr}{1}& {2}& {1}\\ {2}& {4}& {2}\\ {2}& {8}& {1}\end{array}\right]$ (1)
 > $l≔\mathrm{eig}\left(A\right)$
 ${l}{≔}\left[\begin{array}{c}{0}\\ {3}{+}\sqrt{{22}}\\ {3}{-}\sqrt{{22}}\end{array}\right]$ (2)
 > $V,L≔\mathrm{eig}\left(A\right)$
 ${V}{,}{L}{≔}\left[\begin{array}{ccc}\frac{{9}{}\left({3}{+}\sqrt{{22}}\right)}{\left({-}{5}{+}{7}{}\sqrt{{22}}\right){}\left({2}{+}\sqrt{{22}}\right)}& \frac{{9}{}\left({3}{-}\sqrt{{22}}\right)}{\left({-}{5}{-}{7}{}\sqrt{{22}}\right){}\left({2}{-}\sqrt{{22}}\right)}& {-}\frac{{3}}{{2}}\\ \frac{{16}{+}\sqrt{{22}}}{{-}{5}{+}{7}{}\sqrt{{22}}}& \frac{{16}{-}\sqrt{{22}}}{{-}{5}{-}{7}{}\sqrt{{22}}}& \frac{{1}}{{4}}\\ {1}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{ccc}{3}{+}\sqrt{{22}}& {0}& {0}\\ {0}& {3}{-}\sqrt{{22}}& {0}\\ {0}& {0}& {0}\end{array}\right]$ (3)
 > $V,L,N≔\mathrm{eig}\left(A\right)$
 ${V}{,}{L}{,}{N}{≔}\left[\begin{array}{ccc}{-}\frac{{3}}{{2}}& \frac{{9}{}\left({3}{+}\sqrt{{22}}\right)}{\left({-}{5}{+}{7}{}\sqrt{{22}}\right){}\left({2}{+}\sqrt{{22}}\right)}& \frac{{9}{}\left({3}{-}\sqrt{{22}}\right)}{\left({-}{5}{-}{7}{}\sqrt{{22}}\right){}\left({2}{-}\sqrt{{22}}\right)}\\ \frac{{1}}{{4}}& \frac{{16}{+}\sqrt{{22}}}{{-}{5}{+}{7}{}\sqrt{{22}}}& \frac{{16}{-}\sqrt{{22}}}{{-}{5}{-}{7}{}\sqrt{{22}}}\\ {1}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {3}{+}\sqrt{{22}}& {0}\\ {0}& {0}& {3}{-}\sqrt{{22}}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& {2}& {3}\end{array}\right]$ (4)