Examples

I have collected some examples, which illustrates the contents of the database:

First, a matrix must be entered as a square number of integral elements. If the entered matrix is not essential, the essential part is extracted, and this part is used for search in the database. The following four examples makes identical searches of the database:

1 2 3 4

1 2 3 4

1 2 3 4

1 1 2 0 0 0 3 2 4

In the upcoming examples you just should click on the matrices to see the examples.

The following is an example showing, that even if BF_p(A) = BF_p(B), then we can have BF_p²(A) ≠ BF_p²(B). Here p = x − 1:

The following is an example showing, that even for two coprime polynomials p and q with BF_p(A) = BF_p(B) and BF_q(A) = BF_q(B), then we can have BF_pq(A) ≠ BF_pq(B). Here p = x − 1 and q = 2 x + 1:

Even though J^×(A) = J^×(B), we might have BF_p(A) ≠ BF_p(B):

On the other hand we might have BF_p(A) = BF_p(B) but J^×(A) ≠ J^×(B):

A matrix is not necessarily strong shift equivalent to its transposed:

The last is an example of the Baker construction, i.e. a relation between two positive 2×2 matrices with nonnegative determinant, which are similar over Z. In this case the two matrices are each others transposed, and the relation comes in two steps of unit shears, giving a strong shift equivalence of lag 6. Note that the ideal class in the maximal order is not an invariant in this case, but that the maximal order and the equation order are identical, which is what we need.