Trace of a Matrix

To calculate the trace of a square matrix \( M \) of size \( n \), you 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: $$ M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \Rightarrow \mathrm{Tr}(M) = \mathrm{Tr} \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix} $$

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 given scalar c:

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

For \( A^T \) the transposed matrix of A:

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