The different definitions of the signature of a matrix.

linear-algebra matrix-inertia matrix-signature projective-geometry richter-gerbert

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. [1] Nick Higham, “What Is the Inertia of a Matrix?,” Nick Higham. December 2022 [Online]. Available at: [Accessed: February 23, 2023]
  2. [2] Jürgen Richter-Gebert, Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, 1st ed. Springer Publishing Company, Incorporated, 2011.

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