How is the trace of a square matrix defined?

The trace of a square matrix is defined as the sum of its diagonal elements. This definition is fundamental in linear algebra and has several important properties.

To find the trace, you take each of the elements that are positioned along the main diagonal of the matrix (from the top left to the bottom right) and add them together. For a matrix (A = [a_{ij}]) of size (n \times n), the trace is calculated as:

[ \text{Tr}(A) = a_{11} + a_{22} + a_{33} + \ldots + a_{nn} ]

This operation is linear, meaning the trace of the sum of two matrices is equal to the sum of their traces, and it holds under scalar multiplication as well.

The trace has several applications in various fields, including physics and statistics, particularly in the context of linear transformations and determinants, making it a crucial concept in understanding matrix behavior. Other choices such as the product of diagonal elements, differences between eigenvalues, or maximum row sums do not align with the established definition of the trace in linear algebra.