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). So the trace of a square matrix uses these values:

$$ \begin{bmatrix} X & . & . \\ . & X & . \\ . & . & X \end{bmatrix} $$ or, for a rectangular matrix: $$ \begin{bmatrix} X & . & . \\ . & X & . \end{bmatrix} $$ 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: $$ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \\ \mathrm{Tr}(M) = a+d $$

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

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):

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

— The trace is invariant for a cyclic permutation: for A and B of compatible size (and therefore A.B is a square matrix by multiplication">matrix multiplication):

$$ \mathrm{Tr}(AB) = \mathrm{Tr}(BA) \\ \mathrm{Tr}(ABC) = \mathrm{Tr}(CAB) = \mathrm{Tr}(BCA) $$

but in the general case $ \mathrm{Tr}(ABC) \neq \mathrm{Tr}(ACB) \neq \mathrm{Tr}(BAC) $

— For a given scalar c:

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

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

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