How can I show that if the full n-vector space V can be written as a direct sum of subspaces V_i for i=1,... k, such that all V_i are invariant subspaces of diagonalizable matrix A, I can block diagonalize A through a similarity transform
A_blkdiag=(S^-1)AS
where the columns of S contain linearly independent vectors that span first V_1, all the way up to V_k?
I think I got it. Examining A×S, we can note that for the resulting matrix, the column indices corresponding to those of S associated with the vectors spanning V_i can each be written as a linear combination of vectors from V_i, since these subspaces are all invariant subspaces of A. Then, we can rewrite this as A×S=S×D, where D is the matrix containing the coefficients on the linear combinations of the spanning vectors in S. Noting the specified ordering of columns in S, we can note that D is a block diagonal matrix of coefficients. Then, we have (S^-1)×A×S=(S^-1)×S×D=D, and so lemma proved.