Trace of a square matrix

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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)
How to Cite This Entry:
Trace of a square matrix. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article