Eigenvalues

Part of speech: noun

Definitions

1. A specific set of scalar values derived from a matrix that represent the factor by which vectors in a linear transformation are stretched or compressed
2. Distinct numerical values associated with a linear operator which indicate how much, and in which direction, vectors are altered when transformed
3. Numerical values that quantify the magnitude by which directions are scaled when a linear transformation is applied to vectors

Etymology: "Eigenvalues" is the plural of "eigenvalue," a compound of the German-derived prefix "eigen-" (meaning "own" or "characteristic") and "value." The term is a partial calque of the German "Eigenwert," which joins the same two elements: "Eigen" ("own, proper") and "Wert" ("value"). In other words, the word was built to name a value that belongs to a particular operator, a number that is peculiar to it. The idea behind the name is that certain special vectors pass through a transformation without changing direction; they are only stretched or shrunk. The eigenvalue is the factor by which that stretching happens. The prefix "eigen" was added to mark this as a property intrinsic to the operator rather than a value chosen from outside, which is why the word reads as "own value" when taken apart.