I was looking for a definition of the signature of a matrix, and Nick Higham had straightforward definition,

The inertia of a real symmetric \(n \times n\) matrix \(\mathbf{A}\) is a triple, written \(In(\mathbf{A}) = (i_{+}(\mathbf{A}), i_{-}(\mathbf{A}), i_0(\mathbf{A}))\), where \((i_{+}(\mathbf{A})\) is the number of positive eigenvalues of \(\mathbf{A}\), \(i_{-}(\mathbf{A})\) is the number of negative eigenvalues of \(\mathbf{A}\), and \(i_0(\mathbf{A})\) is the number of zero eigenvalues of \(\mathbf{A}\).

The rank of \(\mathbf{A}\) is \(i_{+}(\mathbf{A}) \; + \; i_{-}(\mathbf{A})\). The difference \(i_{+}(\mathbf{A}) \; - \; i_{-}(\mathbf{A})\) is called the signature.

(1, Higham, 2022) (as well as more information about inertia in that post).

I came across the term ‘signature of a matrix’ in (2, Richter-Gebert, 2011), during a discussion of conics. There, the term ‘signature’ seems to be equivalent to Higham’s definition of ‘inertia’.

[This would be a great place for a shrug emoji.]

Cited references

[1] Nick Higham, “What Is the Inertia of a Matrix?,” Nick Higham. December 2022 [Online]. Available at: https://nhigham.com/2022/12/06/what-is-the-inertia-of-a-matrix/. [Accessed: February 23, 2023]

@misc{higham_what_inertia_2022,
title = {What {Is} the {Inertia} of a {Matrix}?},
url = {https://nhigham.com/2022/12/06/what-is-the-inertia-of-a-matrix/},
language = {en},
urldate = {2023-02-23},
journal = {Nick Higham},
author = {Higham, Nick},
month = dec,
year = {2022}
}

The inertia of a real symmetric ..

[2] Jürgen Richter-Gebert, Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, 1st ed. Springer Publishing Company, Incorporated, 2011.

@book{RG_perspectives_2011,
author = {Richter-Gebert, Jürgen},
title = {Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry},
year = {2011},
isbn = {3642172857},
publisher = {Springer Publishing Company, Incorporated},
edition = {1st}
}

Projective geometry is one of the most fundamental and at the same time most beautiful branches of geometry.It can be considered the common foundation of many other geometric disciplines like Euclidean geometry, hyperbolic and elliptic geometry or even relativistic space-time geometry. This book offers a comprehensive introduction to this fascinating field and its applications.In particular, itexplains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplaybetweengeometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians, as well as for computer scientists and physicists. The book is based on the authors experience in implementing geometric software and includes hundreds ofhigh-qualityillustrations.