Describe the set of all matrices C such that $C^{-1}AC = A$

Question

Given a square matrix A, can we find all invertible matrices C such that $A = C^{-1}AC$ ? In other words, can we find a set of all bases such that the matrix of an endomorphism $f$ in those bases is $A$?

The word "find" above should be interpreted as "find a good description/ way of enumeration of the set of such matrices". I believe that one may not be able to do so for all matrices but some of special kinds (such as diagonalizable ones). Anyway I just ask for sure/ to get some insights about this.

To find an "enumeration" seems quite optimistic. – Marc van Leeuwen Sep 20 '13 at 09:12 — Marc van Leeuwen, Sep 20 '13 at 09:12

score 3 · Answer 1 · edited Apr 13 '17 at 12:21

3

$$ A = C^{-1}AC $$ implies $$ CA=AC $$ So you are looking for invertible matrix $C$ such commute with $A$. Look at wiki page on commuting matrices.

Or look as this question.

edited Apr 13 '17 at 12:21

Community

1

answered Sep 19 '13 at 10:43

tom

4,596

2

*invertible matrix C. – vadim123 Sep 19 '13 at 13:38

score 0 · Answer 2 · answered Sep 19 '13 at 10:50

0

Of course, set of all such $C$ always contains the identity matrix. For diagonalizable ones, any scalar multiple of the diagonalizing matrix also belongs to set of $C$.

answered Sep 19 '13 at 10:50

dineshdileep

8,887

Describe the set of all matrices C such that $C^{-1}AC = A$

2 Answers2