spectrum

Politics gives this sense its best-known phrase: good will or opposition across the political spectrum, meaning everyone from one extreme to the other. Continuum and sliding scale are close substitutes. The word also suggests a range of distinct positions arranged in order between two poles. For the plural, English accepts both spectra and spectrums.

Physics names the parts of this range with a modifier: the visible spectrum, the ultraviolet spectrum, the electromagnetic spectrum as a whole. Telecommunications goes further and treats the range as a resource measured in hertz, often with no article, as in "the operator owns spectrum" or "each hertz of spectrum." In that use the word is uncountable, a quantity to be bought and divided rather than a band of colors.

A clipped phrase in which the word autism drops away. It nearly always takes the article, and it most often appears in on the spectrum. The narrator of Percival Everett's Dr. No (2022) says simply "I am on the spectrum." Because autism is left unsaid, the phrase depends on context. Without that context, a reader may hear only the general sense of a range.

Chemistry distinguishes two kinds by the direction the energy travels. An absorption spectrum records what a substance takes in. An emission spectrum records what it gives off once it is heated, irradiated or electrified. Because the pattern belongs to the substance, it serves as an identifying fingerprint. The band of colors is the range itself; here the word means one substance's own pattern within it.

Of all the senses, this one keeps the Latin "image" most literally, and it pairs naturally with the obsolete ghost meaning. One is an image seen where nothing stands; the other is an image left behind once the eyes close. Modern readers know the same experience as an afterimage, the glow that lingers after looking at the sun or a candle flame.

Linear algebra uses the word for a matrix's eigenvalues taken together, usually phrased as the spectrum of A. These values are the roots of the characteristic polynomial. When other kinds of spectra might cause confusion, eigenspectrum is the unambiguous synonym. This is the starting point that the operator sense generalizes: for a matrix, the spectrum and the eigenvalues are simply the same thing.

Functional analysis keeps the word but changes the test. Instead of asking where A − λI sends some nonzero vector to zero, it asks where that operator fails to have a bounded inverse. In infinite dimensions these questions give different answers. For that reason, analysts divide the spectrum into point, continuous and residual parts, and only the point part consists of eigenvalues in the matrix sense.

Algebraic geometers more often write this object than spell it out: Spec(R), or Spec R without parentheses. Its points are prime ideals. The set becomes a geometric object only once the Zariski topology and a structure sheaf of rings are added. With both in place, Spec R is an affine scheme, the building block that takes over the role of the classical algebraic variety.