JCFexample1withCheck.nb

This notebook is on the class web site, on the homework page.

A = {{3,1,1,0},
{-1,1,0,-1},
{-1,-1,0,1},
{-2,-1,-1,1}};
MatrixForm[A]

Eigenvalues[A]

MatrixForm[RowReduce[A - 2 IdentityMatrix[4]]]

A1 = A -  IdentityMatrix[4];
MatrixForm[A1]

MatrixForm[RowReduce[A1]]

MatrixForm[RowReduce[A1.A1]]

Here is the solution:

v1 = {-1,0,1,1};
v2 = {0, -1, 1, 0};
v3 = {0, 0, 0, 1};
v4 = {-1,2,0, 1};

xSoln = v1 Exp[2t]c1 + Exp[t](v2(c2+c3 t) + v3 c3 + v4 c4);
MatrixForm[xSoln]

Now, that form of the general solution is not very pretty, but look how we can check if the solution is correct:

Simplify[A.xSoln - (∂_t) xSoln]

We are now confident that we have solved the problem correctly.  For fun, let us confirm that the Matrix P puts A into Jordan canonical form:

P = Transpose[{v1,v2,v3,v4}];
MatrixForm[P]

The Jordan canonical form of A is

MatrixForm[Inverse[P].A.P]