What does diagonalize mean?
To diagonalize is to convert a square matrix into diagonal form — one in which every entry outside the main diagonal is zero. This is achieved by a change of basis, usually expressed as a similarity transformation using the matrix's eigenvectors. Diagonalization is one of the most valuable operations in linear algebra because a diagonal matrix is far easier to work with: computing its powers, exponentials, or determinants becomes nearly trivial. As a result, the word appears throughout mathematics, physics, and engineering, from solving systems of differential equations to finding the energy states of quantum mechanical systems. The verb derives from "diagonal," which traces back through Latin to Greek roots meaning "from angle to angle." Though strictly technical in register, diagonalize carries a satisfying sense of simplification — of reducing something complicated to its essential, most manageable form.
verbTo convert a square matrix into diagonal form, in which all entries outside the main diagonal are zero, typically by a change of basis. More generally, to express or arrange something along diagonal lines.
- To transform a square matrix into diagonal form by means of a similarity transformation or change of basis.
- Rarely, to arrange or orient something diagonally.
"The spectral theorem guarantees that any real symmetric matrix can be diagonalized by an orthogonal matrix."
"By finding the eigenvalues and eigenvectors, we can diagonalize the matrix and solve the system of differential equations."
"In quantum mechanics, diagonalizing the Hamiltonian reveals the energy eigenstates of the system."
Diagonalizing a matrix doesn't change what it does — it just reveals the hidden axes along which its action is simplest.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of diagonalize
The verb "diagonalize" derives from "diagonal," which entered English in the 16th century via Latin "diagonalis," from Greek "diagonios," meaning "from angle to angle" (from "dia-" meaning "through" or "across," and "gonia" meaning "angle"). The verb form emerged much later, in the context of nineteenth-century developments in linear algebra, when mathematicians sought systematic ways to simplify matrices. It shares its Greek roots with words such as "polygon" and "trigonometry."
Related word forms
How diagonalize is actually used
A technical term used almost exclusively in mathematics, physics, and engineering; it is not part of everyday vocabulary. Both British and American spellings use -ize.
Easily confused with diagonalize
"Diagnose" means to identify the nature of an illness or problem, while "diagonalize" is a mathematical term for converting a matrix to diagonal form.