Matrix trace

• categories: Linear algebra, Definition

Definition
The trace of a square matrix $A\in {\mathbb{R}}^{n\times n}$, denoted $\text{tr}(A)$, is the sum of the elements on its main diagonal:
$\text{tr}(A)={\sum }_{i=1}^n{a}_{ii}.$
For $A=[a_{ij}]$, this is:
$\text{tr}(A)=a_{11}+a_{22}+\cdots +a_{nn}.$

Intuition
The trace provides a simple scalar measure of a matrix’s diagonal elements and often relates to properties of linear transformations, such as their invariant sums under certain transformations.

Key Properties

1. Linearity:
   $\text{tr}(A+B)=\text{tr}(A)+\text{tr}(B),$
   $\text{tr}(cA)=c\cdot \text{tr}(A),$
   for matrices $A,B$ and scalar $c$.

2. Similarity Invariance:
   If $B=P^{-1}AP$ for invertible $P$, then $\text{tr}(B)=\text{tr}(A)$.

3. Cyclic Property (Partial):
   For any matrices $A,B\in {\mathbb{R}}^{n\times n}$,
   $\text{tr}(AB)=\text{tr}(BA).$
   This property extends to cyclic permutations of products but not general reordering.

4. Relation to Eigenvalues:
   The trace of a matrix equals the sum of its eigenvalues (counting algebraic multiplicities):
   $\text{tr}(A)={\sum }_{i=1}^n{\lambda }_i.$

5. Trace of Identity Matrix:
   For $I\in {\mathbb{R}}^{n\times n}$, $\text{tr}(I)=n$.

Applications

• Spectral Analysis: The trace gives the sum of eigenvalues, which is useful in analyzing systems’ dynamics.