Trace¶
Formula¶
\[
\operatorname{tr}(A)=\sum_i A_{ii}
\]
Parameters¶
- \(A\): square matrix
- \(A_{ii}\): diagonal entries
What it means¶
The trace is the sum of diagonal entries of a square matrix.
What it's used for¶
- Matrix identities and proofs.
- Statistics/ML expressions (quadratic forms, covariance identities).
Key properties¶
- Linear: \(\operatorname{tr}(A+B)=\operatorname{tr}(A)+\operatorname{tr}(B)\).
- Cyclic: \(\operatorname{tr}(AB)=\operatorname{tr}(BA)\) when products are defined.
Common gotchas¶
- Cyclic property does not mean arbitrary reordering is valid.
- Trace is defined for square matrices.
Example¶
If \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\), then \(\operatorname{tr}(A)=5\).
How to Compute (Pseudocode)¶
Input: square matrix A (n x n)
Output: trace(A)
s <- 0
for i from 1 to n:
s <- s + A[i,i]
return s
Complexity¶
- Time: \(O(n)\) for an \(n\times n\) matrix when diagonal entries are directly accessible
- Space: \(O(1)\) extra space
- Assumptions: Square matrix input; matrix-loading/storage cost is excluded