In linear algebra, the

**trace**of an

*n*-by-

*n*square matrix

*A*is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of

*A*, i.e.

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

*A*

_{ij}represents the (

*i*,

*j*)'th element of A.

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 |

## Properties

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}).

*A*is an

*n*×

*m*matrix and

*B*is an

*m*×

*n*matrix, then

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

*cyclic property*of the trace. For example, with three square matrices

*A*,

*B*, and

*C*,

- 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*).

*f*:

*V*

`->`

*V*(where

*V*is a finite-dimensional vector space) by choosing a basis for

*V*, describing

*f*as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to similar matrices.

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}.

*A*is always similar to its Jordan form, an upper triangular matrix having λ

_{1},...,λ

_{n}on the main diagonal.

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*)).

## Inner Product

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

*A*= 0. The assignment

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

*m*-by-

*n*matrices.

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}.