Eigenvalues

A matrix A may map some vectors v to multiples of themselves : A.v=cv. In the following examples, v is in green (and is always of unit length), and A.v is in black. When green and black align, an eigenvecotr is found. In Mathematica, all eigenvectors are of unit length.

vec[t_]:={Cos[t],Sin[t]};
o={0,0};

mat={{2,1},{1,2}};

Print[MatrixForm[mat],"   ",MatrixForm[Eigenvectors[mat]],"   ", Eigenvalues[mat]]

Manipulate[Graphics[{Circle[],Green,Thick,Arrow[{o,vec[t]}],Black,Arrow[{o,mat.vec[t]}]},Axes->True,AspectRatio->1,ImageSize->200,PlotRange->{{-2.5,2.5},{-2.5,2.5}},GridLines->Automatic],{t,0,2Pi}]

mat={{.2,1},{1,.2}};
Print[MatrixForm[mat],"   ",MatrixForm[Eigenvectors[mat]],"   ", Eigenvalues[mat]]

Manipulate[Graphics[{Circle[],Green,Thick,Arrow[{o,vec[t]}],Black,Arrow[{o,mat.vec[t]}]},Axes->True,AspectRatio->1,ImageSize->200,PlotRange->{{-2.5,2.5},{-2.5,2.5}},GridLines->Automatic],{t,0,2Pi}]

A 2 D rotation matrix has no real eigenvalues/vectors.

mat={{1,-1},{1,1}};
Print[MatrixForm[mat],"   ",MatrixForm[Eigenvectors[mat]],"   ", Eigenvalues[mat]]

Manipulate[Graphics[{Circle[],Green,Thick,Arrow[{o,vec[t]}],Black,Arrow[{o,mat.vec[t]}]},Axes->True,AspectRatio->1,ImageSize->200,PlotRange->{{-2.5,2.5},{-2.5,2.5}},GridLines->Automatic],{t,0,2Pi}]

For the unit matrix, every vector is an eigenvector.

mat={{1,0},{0,1}};
Print[MatrixForm[mat],"   ",MatrixForm[Eigenvectors[mat]],"   ", Eigenvalues[mat]]

Manipulate[Graphics[{Circle[],Green,Thick,Arrow[{o,vec[t]}],Black,Arrow[{o,mat.vec[t]}]},Axes->True,AspectRatio->1,ImageSize->200,PlotRange->{{-2.5,2.5},{-2.5,2.5}},GridLines->Automatic],{t,0,2Pi}]

3 D matrices

A 3 D rotation matrix (around the z - axis) has one real eigenvector.

alpha=.3;
mat3={{Cos[alpha],Sin[alpha],0},{-Sin[alpha],Cos[alpha],0},{0,0,1}};
Eigenvalues[mat3];
Eigenvectors[mat3];

A more general rotation matrix.

matgen= RotationMatrix[.3,{1,1,1}];
Eigenvalues[matgen];
Eigenvectors[matgen];