Let a and B belong to n-dimensional matrices over complex field, a and B are commutative, that is, ab = ba. It is proved that the eigensubspace of a must be the invariant subspace of B

Let a and B belong to n-dimensional matrices over complex field, a and B are commutative, that is, ab = ba. It is proved that the eigensubspace of a must be the invariant subspace of B

For any vector x belonging to the eigensubspace V λ of eigenvalue λ of A
There is AX = λ X
So a (BX) = Bax = λ BX
So BX belongs to V λ
So the characteristic subspace V λ of a is the invariant subspace of B

Elden Ring Premiere

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