Prove or disprove: Given two diagonalizable matrices A, B that commute (AB = BA),is AB necessarily diagonalizable?
2026-04-03 01:03:20.1775178200
Given two diagonalizable matrices that commute (AB = BA),is AB necessarily diagonalizable?
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Two matrices commutes iff they are diagonalizable simultaneously. This means that $A=PD_1P^{-1}$ and $B=PD_2P^{-1}$, where $D_1,D_2$ are diagonal matrices. This also imply that $AB$ is similar to $D_1D_2$ which is diagonal and then $AB$ is diagonalizable.