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Matrix Trace


The trace of an n×n square matrix A is defined to be

 Tr(A)=sum_(i=1)^na_(ii),
(1)

i.e., the sum of the diagonal elements. The matrix trace is implemented in the Wolfram Language as Tr[list]. In group theory, traces are known as "group characters."

For square matrices A and B, it is true that

Tr(A)=Tr(A^(T))
(2)
Tr(A+B)=Tr(A)+Tr(B)
(3)
Tr(alphaA)=alphaTr(A)
(4)

(Lang 1987, p. 40), where A^(T) denotes the transpose. The trace is also invariant under a similarity transformation

 A^'=BAB^(-1)
(5)

(Lang 1987, p. 64). Since

 (bab^(-1))_(ij)=b_(il)a_(lk)b_(kj)^(-1)
(6)

(where Einstein summation is used here to sum over repeated indices), it follows that

Tr(BAB^(-1))=b_(il)a_(lk)b^(-1)_(ki)
(7)
=(b^(-1)b)_(kl)a_(lk)
(8)
=delta_(kl)a_(lk)
(9)
=a_(kk)
(10)
=Tr(A),
(11)

where delta_(ij) is the Kronecker delta.

The trace of a product of two square matrices is independent of the order of the multiplication since

Tr(AB)=(ab)_(ii)
(12)
=a_(ij)b_(ji)
(13)
=b_(ji)a_(ij)
(14)
=(ba)_(jj)
(15)
=Tr(BA)
(16)

(again using Einstein summation). Therefore, the trace of the commutator of A and B is given by

 Tr([A,B])=Tr(AB)-Tr(BA)=0.
(17)

The trace of a product of three or more square matrices, on the other hand, is invariant only under cyclic permutations of the order of multiplication of the matrices, by a similar argument.

The product of a symmetric and an antisymmetric matrix has zero trace,

 Tr(A_SB_A)=0.
(18)

The value of the trace for a 3×3 nonsingular matrix can be found using the fact that the matrix can always be transformed to a coordinate system where the z-axis lies along the axis of rotation. In the new coordinate system (which is assumed to also have been appropriately rescaled), the matrix is

 A^'=[cosphi sinphi 0; -sinphi cosphi 0; 0 0 1],
(19)

so the trace is

 Tr(A^')=Tr(A)=a_(ii)=1+2cosphi,
(20)

where a_(ii) is interpreted as Einstein summation notation.


See also

Group Character, Matrix, Square Matrix, Tensor Contraction, Tensor Trace

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References

Lang, S. Linear Algebra, 3rd ed. New York: Springer-Verlag, pp. 40 and 64, 1987.Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub.,p. 122, 1993.

Referenced on Wolfram|Alpha

Matrix Trace

Cite this as:

Weisstein, Eric W. "Matrix Trace." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MatrixTrace.html

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