What does diagonalization mean?
Diagonalization is a mathematical term describing two closely related techniques. In linear algebra, it refers to the process of converting a square matrix into diagonal form, so that its only non-zero entries sit on the main diagonal; this simplifies computing powers and exponentials of the matrix and reveals its eigenvalues directly. More broadly, diagonalization names a powerful proof strategy in set theory and logic, made famous by Georg Cantor in 1891: given any purported list of objects, one constructs a new object that differs from each listed item at a corresponding position, thereby showing the list was incomplete. This method underpins results as striking as the uncountability of the real numbers and, through Alan Turing and others, the existence of undecidable problems in computer science. The word belongs strictly to formal academic register, appearing chiefly in textbooks and research literature, and is spelled 'diagonalisation' in British English.
nounThe process of converting a matrix into diagonal form, in which all non-zero entries lie on the main diagonal. More broadly, the technique of proving a statement by constructing an object that differs from every item in a given list along at least one corresponding position, as in Cantor's diagonal argument.
- In linear algebra, the process of transforming a matrix into diagonal form via a similarity transformation.
- In logic and set theory, a proof technique that constructs an object differing from every member of a proposed enumeration, as in Cantor's diagonal argument.
- By extension in computability theory, a self-referential method for showing that certain problems are undecidable.
"Cantor's diagonalization showed that the real numbers cannot be placed in one-to-one correspondence with the natural numbers."
"Because the matrix has distinct eigenvalues, it admits diagonalization over the real numbers."
"Turing's use of diagonalization established that no algorithm can decide, in general, whether an arbitrary program halts."
Rarely used in the plural; when pluralized it refers to distinct instances or methods of diagonalizing different matrices.
"The textbook works through several diagonalizations of increasingly complex matrices."
With a single cleverly constructed list, Georg Cantor used diagonalization in 1891 to prove that some infinities are literally bigger than others.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of diagonalization
Diagonalization derives from 'diagonal', which comes from Latin 'diagonalis', itself from Greek 'diagōnios', meaning 'from angle to angle' (dia-, 'through', and gōnia, 'angle'). The abstract noun formed with the suffix '-ization', borrowed through French and Latin '-ization' from Greek '-izein', arose in nineteenth-century mathematics alongside the development of matrix theory. The same Greek root gōnia appears in cognates such as polygon, trigonometry, and goniometer.
Related word forms
How diagonalization is actually used
Diagonalization is a technical term confined almost entirely to mathematics, logic, and theoretical computer science; it carries no figurative everyday usage. British spelling favours 'diagonalisation'. The word is nearly always used with a definite article or possessive ('the diagonalization of A', 'Cantor's diagonalization') rather than as a count noun.
Easily confused with diagonalization
A diagonal is a straight line joining two non-adjacent corners of a shape, while diagonalization is the mathematical process of reducing a matrix to diagonal form or of constructing a counterexample along a list.