What is the trace of a matrix?

The trace of a matrix is the sum of its diagonal entries.

The trace of a matrix is a scalar value that can be found by adding up all the diagonal entries of the matrix. For example, consider the matrix A:

A =
| 2 3 |
| 4 1 |

The trace of A is 2 + 1 = 3.

The trace of a matrix has several important properties. First, it is invariant under similarity transformations, meaning that if two matrices A and B are similar (i.e. there exists an invertible matrix P such that A = PBP^-1), then they have the same trace. This property is useful in many areas of mathematics, including linear algebra and differential equations.

Another important property of the trace is that it is a linear operator, meaning that if A and B are matrices and c is a scalar, then tr(cA + B) = c tr(A) + tr(B). This property can be proven using the definition of the trace and basic properties of matrix addition and scalar multiplication.

The trace also has a geometric interpretation in terms of the trace of a linear transformation. If T is a linear transformation from R^n to R^n, then the trace of T is the sum of the eigenvalues of T. This interpretation is useful in understanding the behaviour of linear transformations and their relationship to the geometry of the underlying vector space.

In summary, the trace of a matrix is a scalar value that can be found by adding up its diagonal entries. It has several important properties, including invariance under similarity transformations and linearity, and a geometric interpretation in terms of the trace of a linear transformation.