Timeline for When functions commute under composition
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| Nov 23, 2010 at 4:12 | comment | added | Jonas Meyer | One direction of your first statement is clear. In the other direction, after changing bases so that one of the matrices, T, is diagonal, the others will have to be block diagonal (up to conjugating by a permutation matrix) with blocks corresponding to the distinct eigenvalues of T. If they're not yet all diagonal, take a matrix in the set that is not diagonal, and change bases to diagonalize its blocks (which will leave T diagonal because its blocks are scalar). Because the size of the blocks will decrease, this process terminates with the matrices all diagonalized. | |
| Nov 23, 2010 at 3:58 | comment | added | Jonas Meyer | The Gelfand-Neumark theorem you cite generalizes the case where the matrices are normal, and therefore simultaneously unitarily diagonalizable, if you want to think of this as realizing the matrices as complex-valued functions on a finite discrete space. | |
| Nov 23, 2010 at 3:37 | history | answered | Yuval Filmus | CC BY-SA 2.5 |