# How can I get an invertible matrix X with integral entries and AX=XB, where A and B are matrices with integral entries.

Let M be the set of all n by n matrices with integral entries.

For A and B in M, how can I get an invertible matrix X in M with AX=XB in Sage.

I can partially solve that problem in GAP following method.

- V={ X in Mn(QQ) | AX=XB }

(V is a vector space over QQ)

- Basis(V)={B_1,B_2, ..., B_k}

(I wonder which number k it is according to the changes of A and B.)

Make X=a_1 * B_1 + ... +a_k * B_k , a_i in some interval in ZZ.

Check two things which are X in Mn(ZZ) and the existence of the inverse of X in Mn(ZZ).

I can find such X for some easy matrices A,B.

How can I solve this problem in Sage?

Thanks.