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A primer on Vector Calculus.

trace

constituents

for square \(A \in \mathbb{R}^{m\times m}\), we write:

requirements

\(\text{tr}\qty(A) = \sum_{i}^{} A_{ii}\) is the sum of the diagonals

additional information

properties of traces

\begin{equation} \text{tr}\qty(AB) = \text{tr}\qty(BA) \end{equation}

\begin{equation} \text{tr}\qty(ABC) = \text{tr}\qty(CAB) \end{equation}

\begin{equation} \nabla_{A} \qty [\text{tr}\qty(AB)] = B^{T} \end{equation}

As a function of \(n\), what is the runtime of the “worst” input?

• pros: very strong guarantee
• cons: too strong of an upper bound, may have better algorithm for Beyond Worst-Case Analysis