The discussion revolves around a problem from Mary Boas' "Mathematical Methods in the Physical Sciences," specifically regarding the properties of diagonal matrices and their powers. Participants are tasked with demonstrating that if \(D\) is a diagonal matrix, then \(D^{n}\) is also a diagonal matrix with elements equal to the \(n\)th power of the elements of \(D\).
The discussion is ongoing, with participants sharing their thoughts and attempts at formulating a proof. Some have provided hints and suggestions for different approaches, while others are exploring the implications of matrix properties like trace and determinant. There is no explicit consensus yet, but several productive lines of reasoning are being explored.
Participants note their struggles with formal notation and the complexity of summation in matrix operations. There is a recognition of the need for practice and exercises to build confidence in handling such mathematical concepts.
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