- tr(
*A*) =*A*_{1,1}+*A*_{2,2}+ ... +*A*_{n,n}.

If one imagines that the matrix *A* describes a water flow, in the sense that for every **x** in **R**^{n}, the vector *A***x** represents the velocity of the water at the location **x**, then the trace of *A* can be interpreted as follows: given any region *U* in **R**^{n}, the net flow of water out of *U* is given by tr(*A*)· vol(*U*), where vol(*U*) is the volume of *U*. See divergence.

The trace is used to define characters of group representations.

Table of contents |

2 Inner Product 3 Generalization |

The trace is a linear map in the sense that

- tr(
*A + B*) = tr(*A*) + tr(*B*) for all*n*-by-*n*matrices*A*and*B* - tr(
*rA*) =*r*tr(*A*) for all*n*-by-*n*matrices*A*and all scalars*r*.

- tr(
*A*) = tr(*A*^{T}).

- tr(
*AB*) = tr(*BA*).

- tr(
*ABC*) = tr(*CAB*) = tr(*BCA*).

If *A* and *B* are similar, i.e. if there exists an invertible matrix *X* such that *A* = *X*^{-1}*BX*, then by the cyclic property,

- tr(
*A*) = tr(*B*).

There exist matrices which have the same trace but are not similar.

If *A* is a square *n*-by-*n* matrix with real or complex entries and if λ_{1},...,λ_{n} are the (complex) eigenvalues of *A* (listed according to their algebraic multiplicities), then

- tr(
*A*) = ∑ λ_{i}.

From the connection between the trace and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:

- det(exp(
*A*)) = exp(tr(*A*)).

For an *m*-by-*n* matrix *A* with complex (or real) entries, we have

- <
*A*,*B*> = tr(*A*^{*}*B*)

If *m*=*n* then the norm induced by the above inner product is called the Frobenius norm of a square matrix. Indeed it is simply the Euclidean norm if the matrix is considered as a vector of length *n*^{2}.