Trace of a Matrix

The trace of a square matrix is the addition of the values on its main diagonal (starting from the top left corner and shifting one space to the right and down).

$$ \begin{bmatrix} X & . & . \\ . & X & . \\ . & . & X \end{bmatrix} or \begin{bmatrix} X & . & . \\ . & X & . \end{bmatrix} or \begin{bmatrix} X & . \\ . & X \\ . & . \end{bmatrix} $$

To calculate the trace of a square matrix $ M $ of size $ n $, make the sum of diagonal values:

$$ \mathrm{Tr}(M) = \sum_{i=1}^{n} a_{i \, i} $$

- For a 2x2 matrix: $$ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \\ \mathrm{Tr}(M) = a+d $$

Example: $$ M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \\ \mathrm{Tr}(M) = 1+4 = 5 $$

- For a 3x3 matrix: $$ M = \begin{bmatrix} a & b & c \\d & e & f \\ g & h & i \end{bmatrix} \\ \mathrm{Tr}(M) = a+e+i $$

- For rectangular matrix $ M $ of size $ m \times n $, the diagonal used is the one of the included square matrix (from top left corner).

Example: $$ M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \Rightarrow \mathrm{Tr}(M) = \mathrm{Tr} \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix} $$

Calculation from the eigenvalues of a matrix: the trace of a $ M $ matrix is equal to the sum of its eigenvalues (including complex values and multiplicity).

NB: The product of the eigenvalues is the determinant of the matrix.

Trace follows the following properties:

The trace of an identity matrix $ I_n $ (of size $ n $) equals $ n $.

$$ \mathrm{Tr}(I_n) = n $$

For A and B of the same order (that can be added):

$$ \mathrm{Tr}(A + B) = \mathrm{Tr}(A) + \mathrm{Tr}(B) $$

For A and B of compatible size (and therefore A.B is a square matrix by multiplication">matrix multiplication):

$$ \mathrm{Tr}(AB) = \mathrm{Tr}(BA) $$

For a given scalar c:

$$ \mathrm{Tr}(c A) = c \mathrm{Tr}(A) $$

For $ A^T $ the transposed matrix of A:

$$ \mathrm{Tr}(A^T) = \mathrm{Tr}(A) $$