Trace
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Revision 3385
9/1/2026, 9:57:38 AM · araucaria araucana
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The trace of a [[square matrix|square matrix]] is the sum of its diagonal elements.2
3
Let $R$ be a ring and $A = (a_{ij})_{1 \le i,j \le n} \in M_{n}(R)$ be a matrix. The trace $\text{tr}$ is a linear form defined as4
\[5
\text{tr}(A) = \sum_{i=1}^{n}a_{ii}.6
\]7
For example, we have8
\[9
\text{tr}\begin{pmatrix}10
1 & 2 & 3\\11
4 & 5 & 6 \\12
7 & 8 & 913
\end{pmatrix} = 1 + 5 + 9 = 15.14
\]15
### Properties 16
- The trace has a cyclic property ; if $A$ and $B$ are two matrices, then $\text{tr}(AB) = \text{tr}(BA)$.Revision 3383
9/1/2026, 9:39:57 AM · araucaria araucana
Updated text
Compare with revision 26415 changed lines
1
The trace of a [[square matrix|square matrix]] is...1
The trace of a [[square matrix|square matrix]] is the sum of its diagonal elements.2
3
Let $R$ be a ring and $A = (a_{ij})_{1 \le i,j \le n} \in M_{n}(R)$ be a matrix. The trace $\text{tr}$ is a linear form defined as4
\[5
\text{tr}(A) = \sum_{i=1}^{n}a_{ii}.6
\]7
For example, we have8
\[9
\text{tr}\begin{pmatrix}10
1 & 2 & 3\\11
4 & 5 & 6 \\12
7 & 8 & 913
\end{pmatrix} = 1 + 5 + 9 = 15.14
\]Revision 264
7/7/2026, 10:41:05 AM · Ancient Tree
Concept created
The trace of a [[square matrix|square matrix]] is...