autonne

Autonne is primarily known as the surname of the French mathematician Eugène Autonne, whose work contributed to early 20th‑century matrix theory and functional analysis. The name is attached to a canonical result in linear algebra that bears his name alongside Teiji Takagi: the Autonne–Takagi factorization.

The Autonne–Takagi factorization describes how any complex symmetric matrix can be decomposed in a unitary-congruent form. Specifically, for a complex symmetric matrix A, there exists a unitary matrix U and a real nonnegative diagonal matrix Σ such that A = U Σ U^T. The diagonal entries of Σ are the nonnegative singular values of A, and the columns of U are the corresponding singular vectors. This result highlights the special structure of complex symmetric matrices and is analogous in spirit to the spectral theorem for Hermitian matrices, though it uses unitary congruence (A = U Σ U^T) rather than similarity.

In practice, the Autonne–Takagi factorization is used in various areas of applied mathematics, physics, and engineering, including multivariate analysis, quantum information, signal processing, and numerical linear algebra. The factorization is often referred to as the Takagi factorization in some texts, with Autonne’s contribution noted as an essential precursor or co-discovery. The collaboration of Autonne and Takagi underscores the cross-cultural development of linear algebra in the early 20th century.

Biographical information on Eugène Autonne is relatively sparse in standard references, but his name endures in the Autonne–Takagi decomposition. The concept remains a fundamental tool for handling complex symmetric matrices and is routinely taught in advanced linear algebra courses.