Eigenvalues of Matrix

22 March 2000

gives a Matrix M induced from the substitution, and 
	showEigenvalue[M]
shows a graphics of the distribution for these eigenvalues.

Matrix

M={{1,1,1,1},{1,0,0,0},{0,1,0,0},{0,0,1,0}}

M// MatrixForm

M={{1,1,1},{1,0,0},{0,1,0}}

M// MatrixForm

Eigenvalues and its Graphic representation

For a given matrix M , Eigensystem[M] gives 
    { {list of eigenvalues}, {list of eigenvectors} }.
Table[Part[Eigensystem[M], 1, i], {i, Length[lst]}] gives a list of eigenvalues. 
Part[Eigensystem[*], 1, i] gives an i-th eigenvalues, and Part[Eigensystem[*], 2, i] an i-th eigenvector.
showEigenvalue[M_List, lst_List] shows a graphics of the distribution for these eigenvalues.

system = Eigensystem[M] // N

Table[Part[system, 1, i], {i, 3}] (* list of eigenvalues *)

Table[Part[system, 2, i], {i, 3}] (* list of eigenvectors *)

Graphic representation of Eigenvalues

For elist(a list of eigenvalues), showEigenvalue[elist] show thier configurations on Re-Im plane.

listofeigenvalues = Table[Part[system, 1, i], {i,3}]

plotEigenvalue[lst_List] := Module[{x,eigenpoint},
		pt[x_] := Point[{Re[x], Im[x]}];
		eigenpoint = Map[pt[#] &,lst];
		Show[Graphics[{AbsolutePointSize[8],eigenpoint, Circle[{0,0}, 1]}], Axes->Automatic, AspectRatio->Automatic, AxesLabel->{"Re", "Im"}]
		]

plotEigenvalue[listofeigenvalues]

showEigenvalue[M_List] := Module[{eigenpoint},
		pt[x_] := Point[{Re[x], Im[x]}];
		eigenpoint = Map[pt[#] &,Table[Part[Eigensystem[M] // N, 1, i], {i, Length[M]}]];
		Show[Graphics[{AbsolutePointSize[8],eigenpoint, Circle[{0,0}, 1]}], Axes->Automatic, AspectRatio->Automatic, AxesLabel->{"Re", "Im"}]
		]

showEigenvalue[M]

Samples

3-dim

M1={{1,1,1},{1,0,0},{0,1,0}}

M1//MatrixForm

showEigenvalue[M1]

M2={{1,1,1},{1,0,0},{0,1,0}}

M2//MatrixForm

showEigenvalue[M2]

M3={{0,1,1,1},{1,1,1,0},{0,0,1,1},{1,0,0,0}}

M3//MatrixForm

showEigenvalue[M3]

M4={{1,0,1},{1,1,0},{0,1,0}}

M4//MatrixForm

showEigenvalue[M4]

M5={{1,0,1},{0,1,0},{1,1,0}}

M5//MatrixForm

showEigenvalue[M5]

M6={{1,1,1},{1,1,0},{0,1,0}}

M6//MatrixForm

showEigenvalue[M6]

M7={{1,1,0},{1,0,1},{0,1,0}}

M7//MatrixForm

showEigenvalue[M7]