Trace (linear algebra)

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the trace of a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> A, denoted tr(A), is the sum of the elements on its <a href="/facts/Main_diagonal/rkT2F08Q">main diagonal</a>, 
 
 
 
 
 a
 
 11
 
 
 +
 
 a
 
 22
 
 
 +
 ⋯
 +
 
 a
 
 n
 n
 
 
 
 
 {\displaystyle a_{11}+a_{22}+\dots +a_{nn}}
 
. It is only defined for a square matrix (n × n). 
The trace of a matrix is the sum of its <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> (counted with multiplicities). Also, tr(AB) = tr(BA) for any matrices A and B of the same size. Thus, <a href="/facts/Matrix_similarity/ySevpav1">similar matrices</a> have the same trace. As a consequence, one can define the trace of a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> mapping a finite-dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a> into itself, since all matrices describing such an operator with respect to a basis are similar.
The trace is related to the derivative of the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> (see <a href="/facts/Jacobi%27s_formula/RhxPLByr">Jacobi's formula</a>).

Trace (linear algebra) open-in-new

Trace (linear algebra)