The discussion centers on proving the equality Tr(AB) = Tr(BA) for nxn matrices A and B. Participants utilize double sigma notation to express the trace of the product of matrices, specifically ∑p=1n∑k=1nApkBkp and ∑p=1n∑k=1nBpkAkp. The concept of 'dummy indices' is clarified, allowing for the interchange of indices without affecting the validity of the formula.
Students and educators in linear algebra, mathematicians interested in matrix properties, and anyone looking to deepen their understanding of matrix operations and proofs.
Bipolarity said:Homework Statement
Prove the following statement, where A and B are nxn matrices.
Tr(AB) = Tr(BA)
Homework Equations
The Attempt at a Solution
Using some manipulations, I arrived at
\sum^{n}_{p=1}\sum^{n}_{k=1}A_{pk}B_{kp} = \sum^{n}_{p=1}\sum^{n}_{k=1}B_{pk}A_{kp}
If I can prove the above, I am done, but I have never worked with double sigma notations before, so any advice folks?
BiP
Dick said:k and p are what are called 'dummy indices'. You can change k to any other letter, like q and you still have a good formula for Tr. Just interchange k and p.