“Trace (Matrix)”的概念、定义、习题解题方法及翻译-数学辞典

单词

Trace (Matrix)

释义

Trace (Matrix)

The trace of an Square Matrix A is defined by

(1)

For Square Matrices A and B, it is true that

			(2)
			(3)
			(4)

(Lange 1987, p. 40). The trace is invariant under a Similarity Transformation

(5)

(Lange 1987, p. 64). Since

(6)



			(7)

where

is the Kronecker Delta.

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


			(8)

Therefore, the trace of the Commutator of

and

is given by

(9)

The product of a Symmetric and an Antisymmetric Matrix has zero trace,

(10)

The value of the trace can be found using the fact that the matrix can always be transformed to a coordinate system where thez-Axis lies along the axis of rotation. In the new coordinate system, the Matrix is

(11)

so the trace is

(12)

References

Lang, S. Linear Algebra, 3rd ed. New York: Springer-Verlag, pp. 40 and 64, 1987.

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