What does diagonalise mean?
To diagonalise a matrix is to transform it — through a carefully chosen change of basis — into a simpler form whose only nonzero entries lie on the main diagonal. This is one of the most powerful techniques in linear algebra: once a matrix has been diagonalised, difficult problems such as computing powers of the matrix, solving systems of differential equations, or finding eigenvalues become straightforward. The procedure relies on the matrix's eigenvectors, which serve as the natural coordinate directions along which the transformation acts independently. The term appears almost exclusively in mathematical, physical, and engineering contexts; physicists routinely 'diagonalise the Hamiltonian', for instance, to extract a quantum system's energy levels. Spelled 'diagonalise' in British English and 'diagonalize' in American English, the word derives from 'diagonal', which traces back through Latin to Greek roots meaning 'from angle to angle'.
verbTo convert a square matrix into diagonal form, in which all entries outside the main diagonal are zero, typically by a change of basis or similarity transformation.
- (mathematics) To convert a square matrix into diagonal form via a similarity transformation, so that all non-diagonal entries become zero.
"By choosing the eigenvectors as a new basis, we can diagonalise the matrix and read off its eigenvalues directly."
"The physicist showed how to diagonalise the Hamiltonian to find the system's energy levels."
"Since the two matrices commute, they can be diagonalised simultaneously by the same change of basis."
Every time you let Google rank pages or Netflix recommend a film, some hidden matrix was probably diagonalised first — it's one of mathematics' great simplifying moves.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of diagonalise
The verb 'diagonalise' is formed from 'diagonal' plus the verb-forming suffix '-ise'. 'Diagonal' itself entered English from Latin 'diagonalis', derived from Greek 'diagonios', meaning 'from angle to angle', formed from 'dia-' ('through, across') and 'gonia' ('angle, corner'). Related words sharing this Greek root for angle include 'polygon' ('many angles'), 'trigonometry' ('triangle measurement'), and 'diagonally'. The technical mathematical sense arose in the nineteenth century alongside the modern development of linear algebra and matrix theory.
Related word forms
How diagonalise is actually used
This spelling with '-ise' is standard in British English; American English uses 'diagonalize'. The word belongs almost exclusively to mathematical, scientific, and technical registers and would rarely appear in everyday conversation.
Easily confused with diagonalise
'Diagonal' is a noun or adjective referring to a slanting line across a shape, while 'diagonalise' is the verb meaning to transform a matrix into diagonal form.