Trace

Trace has several related senses. It can mean a mark left by something, a very small amount of something, or the act of following evidence to locate something. As a verb, trace can mean to draw or copy by following lines, or to discover the origin or path of something. The word comes from Latin trahere via Old French trace.

In mathematics, the trace of a square matrix is the sum of its diagonal elements. It is linear and invariant under similarity transformations; it equals the sum of eigenvalues. Properties include tr(A+B) = tr(A) + tr(B) and tr(kA) = k tr(A). It is not generally multiplicative: tr(AB) does not equal tr(A) tr(B).

In science, trace denotes very small quantities, such as trace elements or trace gases. Trace analysis seeks to measure minuscule amounts using sensitive instruments like atomic absorption spectroscopy, ICP-MS, or gas chromatography.

In computing, tracing records information about program execution or system behavior, used for debugging and performance optimization. Traces or trace logs capture events across software or hardware. Tools include strace and DTrace, and tracing is also used in networking via traceroute to map the path to a destination.

Other uses include trace fossils, which are preserved marks of an organism's activity, and the general sense of a visible or detectable mark left behind.