1

Is there a way to determine all the matrices that commute with a given, say real, matrix? Or, can we say something more in some particular cases (symmetric matrices...)?

user26857
  • 52,094
Vladimir
  • 45

1 Answers1

1

the extreme case is when the minimal polynomial of the matrix coincides with its characteristic polynomial. That is, each eigenvalue occurs in just one Jordan block. When this happens, the only matrices commuting with it are polynomials $a_0 I + a_1 M + a_2 M^2 + \cdots + a_{n-1} M^{n-1.} $ There is no need to use higher exponents, as $M$ obeys its own characteristic polynomial. This set, as a vector space, has dimension $n.$

The other extreme is the scalar multiples of the identity matrix, all matrices commute, the dimension is $n^2.$

Will Jagy
  • 139,541
  • Thx. Both cases can be found in many places. I believe that, in general, the situation is complicated, but I hope that there are some results. – Vladimir Nov 22 '19 at 23:20
  • @Vladimir see https://math.stackexchange.com/questions/1476010/computing-the-dimension-of-a-vector-space-of-matrices-that-commute-with-a-given and the more general Cecioni Frobenius http://www.numbertheory.org/courses/MP274/applic.pdf – Will Jagy Nov 23 '19 at 00:44