What is the trace of a matrix?

The trace of a square matrix is the sum of the elements on its main diagonal. In other words, it is the sum of the elements a_11, a_22, a_33, ..., a_nn of the matrix A, where n is the size of the matrix.

The trace of a matrix is denoted by tr(A) or Tr(A). The trace of a matrix is a scalar quantity and is invariant under similarity transformations, meaning that if two matrices A and B are similar (i.e., B = P^-1AP for some invertible matrix P), then tr(A) = tr(B).

The trace of a matrix has several important properties, such as being linear, meaning that tr(A + B) = tr(A) + tr(B) and tr(kA) = k tr(A) for any scalar k, and being cyclical, meaning that tr(ABC) = tr(BCA) = tr(CAB) for square matrices A, B, and C of compatible sizes.

The trace of a matrix is commonly used in various areas of mathematics and physics, such as in calculations involving eigenvalues, determinants, and matrix norms.