# 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] N. 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]
2. [2] J. 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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