What is the trace of a matrix?

The trace of a square matrix is the sum of the elements on its main diagonal. The main diagonal comprises elements for which the row index and column index are the same. Mathematically, the trace of an n x n matrix, denoted as A, can be represented by:

$$\text{Tr}(A) = \sum_{i=1}^{n} a_{ii}$$

Here, a_ii indicates the element of matrix A in the ith row and ith column.

For example, given a matrix A:

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

The trace of this 3 x 3 matrix would be:

Tr(A) = a₁₁ + a₂₂ + a₃₃

It’s important to note that the trace is only defined for square matrices (i.e., matrices with an equal number of rows and columns). Also, keep in mind that the trace of a matrix has several interesting properties, such as:

1. The trace of a matrix is equal to the sum of its eigenvalues.

2. The trace is invariant under similar transformations, meaning that if A and B are similar matrices (i.e., B = P^− 1AP for some invertible matrix P), then their traces are equal.

3. The trace is a linear operation, which implies that for any matrices A and B, and scalar α, we have:
Tr(A + B) = Tr(A) + Tr(B),   Tr(αA) = αTr(A)