Definition:Trace (Linear Algebra)

From ProofWiki
Jump to navigation Jump to search

This page is about traces in the context of linear algebra. For other uses, see Definition:Trace.

Definition

Matrix

Let $A = \sqbrk {a_{ij} }_{1 \mathop \le i, j \mathop \le n}$ be a matrix.


The trace of $A$ is:

$\displaystyle \tr \paren A = \sum_{i \mathop = 1}^n a_{ii}$


Linear Operator

Let $V$ be a vector space.

Let $A: V \to V$ be a linear operator of $V$.


The trace of $A$ is the trace of the matrix of $A$ with respect to some basis.