JCFexample2withCheck.nb

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

A = {{0,1,1,4},
{-1,0,2,-1},
{0,0,1,-1},
{0,0,2,-1}};
MatrixForm[A]

Eigenvalues[A]

AI = A - I IdentityMatrix[4];
MatrixForm[AI]

MatrixForm[RowReduce[AI]]

MatrixForm[RowReduce[AI.AI]]

Now do the computations on paper, using these results.

v1 = {1, 11, 0, 0};
u1 = {11, -1, 0, 0};
v2 = {-7, 0, 2, 0};
u2 = {1, 0, 2, 4};

Put in the solution to x dot = A x and check it.

xSoln = v1(r1 Cos[t + θ1] + r2 t Cos[t+θ2])+
 u1 (r1 Sin[t + θ1] + r2 t Sin[t+θ2])+
v2 r2 Cos[t+θ2] + u2 r2 Sin[t+θ2]

Simplify[A.xSoln - (∂_t) xSoln]

P = Transpose[{u1+I v1, u2+I v2, u1 - I v1, u2-I v2}];
MatrixForm[P]

The Jordan canonical form of A is:

MatrixForm[Inverse[P].A.P]

Now, I will get the book's “real” Jordan canonical form of A:

Preal = Transpose[{v1, u1, v2, u2}];
MatrixForm[Preal]

MatrixForm[Inverse[Preal].A.Preal]

Just for fun, let's look at the what Mathematica can do.

MatrixExp[A t]

Yeech.  To make it look (somewhat) civilized, I had to do these various commands.  Note the use of the "at" symbol for composition of functions in Mathematica.  It is very useful since it avoids having to do

MatrixForm[Simplify[ComplexExpand[MatrixExp[A t]]]]

which has all those "]"s at the end.

MatrixForm@Simplify@ComplexExpand@MatrixExp[A t]