# Definition The *trace* of a [[square matrix]] is defined by $\text{tr}(A) = \sum_{i=1}^n a_{ii}$ It has the following properties: $ \begin{align} \text{tr}(B^{-1} A B) &= \text{tr}(A),\\ \text{tr}(A) &= \lambda_1 +\lambda_2 +\ldots +\lambda_n \end{align} $ where $\lambda_i$ are the eigenvalues of $A$. The first property says that it is invariant under [[Similarity Transformation|similarity transformations]]. Moreover, we also have: $ \det(e^X) = e^{\text{tr}(X)} $ where $\det$ denotes the [[determinant]]. This identity is proved [[Trace and Determinant as Homomorphisms|here]]. # As a Functional The trace is a [[functional]] on the [[vector space]] $M_n(\mathbb{F})$ (see [[matrix|here]] for a definition of this vector space).