The discussion centers on proving that if \(D\) is a diagonal matrix, then \(D^n\) is also a diagonal matrix with elements equal to the nth power of the elements of \(D\). Participants suggest using index notation and induction as effective methods for the proof. Key points include the observation that the trace and determinant do not suffice to establish the equality of matrices, and the necessity of formalizing the argument using proper summation notation. The conversation highlights the importance of clarity in mathematical proofs, particularly when dealing with indices.
PREREQUISITESStudents and educators in mathematics, particularly those studying linear algebra and matrix theory, as well as anyone seeking to strengthen their proof-writing skills in mathematical contexts.
kq6up said: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