The trace of an square matrix is defined to be
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(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 and , it is true that
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(2) | |||
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(3) | |||
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(4) |
(Lang 1987, p. 40), where denotes the transpose. The trace is also invariant under a similarity transformation
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(5) |
(Lang 1987, p. 64). Since
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(6) |
(where Einstein summation is used here to sum over repeated indices), it follows that
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(7) | |||
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(8) | |||
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(9) | |||
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(10) | |||
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(11) |
where is the Kronecker delta.
The trace of a product of two square matrices is independent of the order of the multiplication since
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(12) | |||
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(13) | |||
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(14) | |||
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(15) | |||
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(16) |
(again using Einstein summation). Therefore, the trace of the commutator of and is given by
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(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,
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(18) |
The value of the trace for a 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
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(19) |
so the trace is
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(20) |
where 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
Cite this as:
Weisstein, Eric W. "Matrix Trace." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MatrixTrace.html