Notes - Linear Algebra II HT23, Trace
Flashcards
What is the trace $\text{Tr}(A)$ of a square matrix $A$?
The sum of the diagonal entries.
What is the trace $\text{Tr}(T)$ of a linear transformation $T$?
The sum of the diagonal entries of a matrix $A$ representing $T$ with respect to some basis.
What’s a quick proof that the trace $\text{Tr}(T)$ of a linear transformation $T$ is the same for any choice of basis?
Consider $A, B$ both representing $T$ with respect to different bases
\[\begin{aligned} \text{Tr}(B) &= \text{Tr}(P^{-1} A P) \\\\ &= \text{Tr}(P^{-1}P A) \\\\ &= \text{Tr}(IA) \\\\ &= \text{Tr}(A) \end{aligned}\]Proofs
Prove that $\text{Tr}(AB) = \text{Tr}(BA)$.
Todo.