Let A be a square matrix over $\mathbb{C} $ and let B be its Jordan canonical form then are A and B similar over a proper subfield of $\mathbb{C}$?

Question

Let A be an $ n\times n $ matrix over $\mathbb {C} $ and let B be its Jordan canonical form , then A and B are similar over $\mathbb {C} $. Now suppose K be a proper subfield of $\mathbb {C} $ , in general A and B are not similar over K .

But If the entries of A also belong to K and the characteristic polynomial of A splits over K , then can I say that A and B are similar over K?

Actually, I am supposed to find a condition so that A and B would be similar over any proper subfield of $\mathbb {C} $.

Any help would be appreciated. Thanks in advance.

score 2 · Accepted Answer · answered May 24 '19 at 09:20

If $K$ is a subfield of $\mathbb{C}$ which contains the entries of $A$ and $B$, then $A$ and $B$ are guranteed to be similar on the smaller field $K$. You do not need the hypothesis of the characteristic polynomial splitting over $K$ (if you assume this, you can give a proof by using the fact that $A$ will have a Jordan canonical form $J_{K}$ over $K$ and then showing that $J_K$ and $B$ only differ by a permutation of the blocks) Otherwise, in the general case, you can make use of the Rational canonical form; refer to the answer in this post about similar matrices and field extensions

Thanks a lot sir,very helpful – suchanda adhikari May 24 '19 at 23:10 — suchanda adhikari, May 24 '19 at 23:10

Let A be a square matrix over $\mathbb{C} $ and let B be its Jordan canonical form then are A and B similar over a proper subfield of $\mathbb{C}$?

1 Answers1