Power of a Diagonalized Matrix?

[kq6up]

Yes, I see. The sum is gone altogether, so $$ { \left[ { { A }^{ 2 } } \right] }_{ ij }={ A }_{ ii }{ A }_{ ii }\quad When\quad i=j,\quad and\quad { \left[ { { A }^{ 2 } } \right] }_{ ij }=0\quad when\quad i\neq j $$.

Chris

 

[Dick]

Yes, I see. The sum is gone altogether, so $$ { \left[ { { A }^{ 2 } } \right] }_{ ij }={ A }_{ ii }{ A }_{ ii }\quad When\quad i=j,\quad and\quad { \left[ { { A }^{ 2 } } \right] }_{ ij }=0\quad when\quad i\neq j $$.

Chris

That's it! The nth power goes exactly the same way. There is at most one nonzero term in that big summation.