Eigenletters

Eigenletters are an extraction method used in hunt puzzles, wherein two words or phrases with identical lengths share one or more letters in the same position(s).

Etymology
The word "eigenletter" is a play on the concepts of eigenvalues and eigenvectors in linear algebra (particularly the former). While the algebraic term refer to vectors that do not change direction when linear transformations are applied to them, and the amount by which those vectors stretch or contract, eigenletters have a less complicated relationship with their component words.

The closest comparison to the original eigenvalue concept that can be made is that one's answer words are comparable to vectors. Sometimes, their content (true value) changes, but their length (direction) stays the same. In these cases, the eigenletters are the amount that these answers stayed the same, rather than the amount they've changed (as is the case with eigenvalues).

The coinage of the term "eigenletter" traces to Teammate Hunt 2020.

Puzzle Applications
Eigenletters as traditionally defined requires identical-length answers sharing one or more letters in the same position. This extraction may also require reordering of the extracted letters, which may be done by such methods as alphabetization (if words share starting letters or have consecutive starting letters), or ordering by length, as seen below. PAWN JACK QUEEN CROWN SHIELD SWORDS BOWLING LAWYERS VARIABLE SYNDROME ALGORITHM INTERFACE When pairs of phrases with more than one word have relevant eigenletters, they usually share their punctuation and spacing along with their length.

Strategy
While the actual application of eigenletters for extraction is easy, the tough part is identifying when it's meant to be used. If one finds that they have an even number of answers (words or phrases), and they can be paired up into equal lengths (usually ignoring spaces), it's a good indicator that eigenletters should be looked at.

However, a wrench gets thrown into the mix when eigenletters are attempted to be used with words/phrases of differing length. In these cases, unless it's sufficiently clued (either by flavortext or by direct instruction), it's near impossible to determine with certainty that it's meant to be used for extraction (and also goes against the meaning of the name). Thankfully, these niche cases aren't very common, and most cases of this type of extraction that one will encounter will be squarely in the former category.