Problem statement: Let be a matrix made up of two diagonal blocks: . Then is diagonalizable if and only if and are diagonalizable.
Solution: Let be matrices. Then is an matrix. Let be the linear operator and be the basis of the vector space w.r.t the matrix . Denote by W and by W’. Then W and W’ are T-invariant subspaces. Also, is the direct sum of and .
The if part:
Let be diagonalizable. Then there is a basis of eigenvectors of the vector space . Let the basis be Then for some and . Then .
Then . Since W and W’ are T-invariant, . Then the nonzero and are eigenvectors of W and W’ respectively. Also, since V is the direct sum of W and W’, and is a basis of V, generates W and generates W’. Then we can obtain a basis of eigenvectors for W and W’ by dropping the dependent vectors in and . This essentially means that A and D are diagonalizable.
The only-if part:
As are diagonalizable, let and be diagonal matrices where are invertible. Then diagonalizes .