Trace

The trace of a square $n \times n$ matrix is the sum of the elements of the main diagonal of the matrix. For example, the trace of the matrix $M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}$ is $\text{tr}(M) = 1 + 5 + 9 = 15$ .

Properties

Viewed as a function from $n \times n$ matrices to the underlying field (frequently the real numbers), the trace is a linear map: it is not difficult to verify that $\text{tr}(cA + B) = c\, \text{tr}(A) + \text{tr}(B)$ for any $n \times n$ matrices $A$ and $B$ and any scalar $c$ .

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Trace

Properties

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