Trace of a square matrix
The sum of the entries on the main diagonal of this matrix. The trace of a matrix is denoted by , or :
Let be a square matrix of order over a field . The trace of coincides with the sum of the roots of the characteristic polynomial of . If is a field of characteristic 0, then the traces uniquely determine the characteristic polynomial of . In particular, is nilpotent if and only if for all .
If and are square matrices of the same order over , and , then
while if ,
The trace of the tensor (Kronecker) product of square matrices over a field is equal to the product of the traces of the factors.
|[a1]||P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 336|
|[a2]||F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1959) (Translated from Russian)|
Trace of a square matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trace_of_a_square_matrix&oldid=16032