UltimateJordan.mws

> with(linalg):

```Warning, the protected names norm and trace have been redefined and unprotected
```

> A:=matrix([[0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0], [1, 0, 0, 2, 0, 0, 0], [0, 0, -1, 0, 2, 0, 0], [-3, 2, 0, -1, 1, 2, 1], [0, -1, 0, 0, 0, 0, 1]]);

> #The Jordan form of this matrix can be found using#

> jordan(A);

> #Note that Maple is always irritating. The blocks are not in the natural order.... We will now explain how to find the Jordan form.

> #We start by calculating the minimal polynomial. There are several methods we discussed that give that but here we will just use

> factor(charpoly(A,t));factor(minpoly(A,t));

> #Note that this alone doesn't determine the Jordan form. There could be either a 2,2 or a 2,1,1 configuration of Jordan blocks for the eigenvalue 1. We know that R^7 = W1 directsum W2 where W1 is the kernel of (A-1)^2 and W2 is the kernel of (A-2)^3. Note that to calculate the kernels of U1:=A-1 and its powers on W1 is like calculating the same thing on the total space and that's what we do first:

> B:=kernel(A-1);

> Ba:=kernel((A-1)^2);

> #This is already W1 and we need not go further. Note that we now know that the configuration of the Jordan blocks is 2,2. Now for W2:

> kernel(A-2);

> kernel((A-2)^2);

> kernel((A-2)^3);

> #This is already W2 and we need not go further.

> #We want to find a good basis for the nilpotent operator U1 defined as A-1 restricted to W1. The first step is to find a complement to kernel(A-1) inside kernel((A-1)^2). We called that complement C2. Note that Steinitz substitution lemma guarantees that if we take the basis B to kernel(A-1) given above then there are two vectors among the basis of kernel((A-1)^2) given above that will complete B to a basis of kernel((A-1)^2). We guess that these could be Ba[1] and Ba[2]. We check the rank of

> M:=matrix([B[1], B[2], Ba[1], Ba[2]]);rank(M);

> #So our guess is true. Now forget about B and just remember the vectors Ba[1], Ba[2]

> v1 := vector([0, 1, 0, 0, 0, -1, 0]); v2 := vector([0, 0, 0, 1, 0, -1, 0]);

> #The next step is to apply (A-1) to these vectors

> v3:=multiply(A-1,v1); v4:=multiply(A-1, v2);

> #Note that v3 and v4 are a basis to kernel of A-1 and we are done with W1. Now for W2:

> Bb:=kernel(A-2);

> Bc:=kernel((A-2)^2);

> Bd:=kernel((A-2)^3);

> #We need to decompose kernel(A-2)^3 as kernel((A-2)^2) directsum C3. Clearly we can let C3 be the span of Bd[2]. So let

> v5:=Bd[2];

> #We calculate now

> v6:=multiply(A-2, v5);

> #In principle we are looking for a decomposition kernel(A-2)^2 = kernel(A-2) directsum C2 where C2 contains v6. Dimension considerations show that C2 is just the span of v6. Let

> v7:=multiply(A-2, v6);

> #I claim that in the basis v3, v1, v4, v2, v7, v6, v5 the matrix has the form

> matrix([[1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 2, 1, 0], [0, 0, 0, 0, 0, 2, 1], [0, 0, 0, 0, 0, 0, 2]]);

> #Let us check that. Let

> M:=transpose(matrix([v3, v1, v4, v2, v7, v6, v5]));

> multiply(inverse(M), A, M);

>