The Trace of a Matrix
In the realm of linear algebra, the trace of a square matrix is the sum of the elements positioned along the main diagonal.
It's key to remember that the concept of trace is only defined for square matrices - that is, matrices with an equal number of rows and columns.
Let's illustrate this with a concrete example.
Suppose you're given the following 3x3 matrix
$$ A = \begin{pmatrix} 4 & 2 & 1 \\ 1 & 5 & -1 \\ 7 & 2 & 3 \end{pmatrix} $$
The trace of matrix A is calculated by summing up the elements on the main diagonal: 4+5+3
$$ Tr(A) = 4+5+3 = 12 $$
Generally, the trace of a matrix is denoted by the symbol "tr(A)" or "Tr(A)".
This is an algebraic sum, which means that when computing a trace, you subtract any negative elements. For instance, given this matrix $$ B = \begin{pmatrix} 4 & 2 & 1 \\ 1 & 5 & -1 \\ 7 & 2 & \color{red}{-3} \end{pmatrix} $$ The trace calculation amounts to 6. $$ Tr(A) = 4+5+(-3) = 9-3=6 $$
Properties of the Trace
The trace of a matrix upholds several properties:
- The trace of the sum of two square matrices A and B equals the sum of the traces of the two matrices $$ Tr(A+B)=Tr(A)+Tr(B) $$
- The trace of a matrix multiplied by a scalar number k is the same as the product of the matrix's trace and the scalar k $$ Tr(k \cdot A) = k \cdot Tr(A) $$
- The trace of the product of two square matrices A·B equals the trace of the product B·A with the order of factors reversed $$ Tr(A \cdot B)=Tr(B \cdot A) $$
- The trace of a matrix A equals the trace of the transpose of matrix A, denoted by AT $$ Tr(A) = Tr(A^T) $$
