Eigensystem Basics

The n x n matrix A will take an n-vector x and map it into an n-vector y,

A x = y

Here is a picture of this mapping in two dimensions for a specific case. The red vector is the original vector, and the blue vector is the vector that is formed by multiplying by the matrix A.

n=2;
A={{1,-1},{-1,-1/10}};
MatrixForm[A]
vec[1]={1,4};
MatrixForm[vec[1]]
MatrixForm[A.vec[1]]

My little set of commands to produce the plot:

evecplot[p_]:=Show[Graphics[{Red,Thick,Arrow[{{0,0},vec[p]}]}],
Graphics[{Blue,Thick,Arrow[{{0,0},A.vec[p]}]}],Axes->True,AxesLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},AspectRatio->1]

Show[evecplot[1],PlotRange->{{-5,5},{-5,5}},ImageSize->{500,Automatic},BaseStyle->20]

For some vectors x, this mapping will produce a vector y which is parallel to the vector x, but scaled by some amount λ,

A x = λ x

Here is a picture of this situation:

n=2;
vec[1]={(1/20) (-11+(Sqrt[521])),1}
vec[2]={(1/20) (-11-(Sqrt[521])),1}

Show[{evecplot[1],evecplot[2]},PlotRange->{{-3,3},{-4,4}},ImageSize->{500,Automatic}]

The vectors for which this is true are called eigenvectors, and each eigenvector has an associated scaling parameter λ, which is the eigenvalue. We can rewrite our system of equations as follows

A x = λ x
A x - λ x = 0
(A  - λ I)x = 0

For this system to have a nonzero solution x, we require the determinant of the matrix A-λ I to be zero. We can solve the following equation for (λ^((k))), k=1,2,3,...,n.

det(A  - (λ^((k))) I) = 0

Once the values of  (λ^((k))) are known, we can solve the equations

(A  - (λ^((k))) I)(x^((k))) = 0

for the eigenvectors (x^((k))).

Mathematica Commands for Eigensystems

We will be solving most of the eigensystems by hand, but if you want to check your work or have a more complicated example in (ℝ^3) that would be difficult to solve by hand you can use Mathematica.

Example 1

A={{1,-1},{-1,-1/10}};
MatrixForm[A]

Here is how you can get a list of the eigenvalues:

evals=Eigenvalues[A]
evals[[1]]
evals[[2]]

Here is how you can get a list of the eigenvectors:

evecs=Eigenvectors[A]
MatrixForm[evecs[[1]]]
MatrixForm[evecs[[2]]]

You can also get them all together, but I don’t find this as useful.

Eigensystem[A]

Check:

A.evecs[[1]]==evals[[1]]evecs[[1]]
A.evecs[[2]]==evals[[2]]evecs[[2]]

Example 2

A={{7,-4},{5,-2}};
MatrixForm[A]

Here is how you can get a list of the eigenvalues:

evals=Eigenvalues[A]
evals[[1]]
evals[[2]]

Here is how you can get a list of the eigenvectors:

evecs=Eigenvectors[A]
MatrixForm[evecs[[1]]]
MatrixForm[evecs[[2]]]

Check:

A.evecs[[1]]==evals[[1]]evecs[[1]]
A.evecs[[2]]==evals[[2]]evecs[[2]]

Example 3

A={{3,-5},{1,-1}};
MatrixForm[A]

Here is how you can get a list of the eigenvalues:

evals=Eigenvalues[A]
evals[[1]]
evals[[2]]

Here is how you can get a list of the eigenvectors:

evecs=Eigenvectors[A]
MatrixForm[evecs[[1]]]
MatrixForm[evecs[[2]]]

Check:

A.evecs[[1]]==evals[[1]]evecs[[1]]
A.evecs[[2]]==evals[[2]]evecs[[2]]

Example 4 in (ℝ^3)

A={{1,-2,2},{-2,1,-2},{2,-2,2}};
MatrixForm[A]

Here is how you can get a list of the eigenvalues:

evals=Eigenvalues[A]
evals[[1]]
evals[[2]]
evals[[3]]

Here is how you can get a list of the eigenvectors:

evecs=Eigenvectors[A]
MatrixForm[evecs[[1]]]
MatrixForm[evecs[[2]]]
MatrixForm[evecs[[3]]]

Check:

A.evecs[[1]]==evals[[1]]evecs[[1]]
A.evecs[[2]]==evals[[2]]evecs[[2]]
A.evecs[[3]]==evals[[3]]evecs[[3]]