Matrix Trace

Definition

if $A\in {\mathrm{M}}_n(F)$, then the trace of matrix $A$ is defined as

$$\mathrm{tr}A=\sum _i^n{A}_{ii}$$

Properties

Sum of eigenvalues

The trace of a matrix is always the sum of its eigenvalues (including repeated ones). This relationship holds because the trace is invariant under similarity transformations and the characteristic polynomial of the matrix (which has roots equal to the eigenvalues) has coefficients that relate to the trace and determinant of the matrix.

Cyclic property

The trace of a matrix is an invariant under the commutation of matrix multiplication

$$\mathrm{tr}(AB)=\mathrm{tr}(BA)$$

Similarly,

$$\mathrm{tr}(ABC)=\mathrm{tr}(BCA)=\mathrm{tr}(CAB)$$

Trace of a product

If $A,B\in F^{m\times n}$, then:

$$\mathrm{tr}(A^{T}B)=\mathrm{tr}(A{B}^T)=\mathrm{tr}(B^{T}A)=\mathrm{tr}(B{A}^T)=\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}{b}_{ij}$$

In particular, if $A,B\in {\mathrm{M}}_n(F)$, we have

$$\mathrm{tr}(AB)=\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}{b}_{ji}=\sum _{i=1}^n\sum _{j=1}^n{a}_{ji}{b}_{ij}$$