Sylvester's law of inertia

Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if $A$ is the symmetric matrix that defines the quadratic form, and $S$ is any invertible matrix such that $D=SAS^{\mathrm {T} }$ is diagonal, then the number of negative elements in the diagonal of $D$ is always the same, for all such ⁠ $S$ ⁠; and the same goes for the number of positive elements.

This property is named after James Joseph Sylvester who published its proof in 1852.^[1]^[2]

^ Sylvester, James Joseph (1852). "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares" (PDF). Philosophical Magazine. 4th Series. 4 (23): 138–142. doi:10.1080/14786445208647087. Retrieved 2008-06-27.
^ Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. pp. 360–361. ISBN 978-0-19-853248-4.

[syl852-1] Sylvester, James Joseph (1852). "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares" (PDF). Philosophical Magazine. 4th Series. 4 (23): 138–142. doi:10.1080/14786445208647087. Retrieved 2008-06-27.

[norm-2] Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. pp. 360–361. ISBN 978-0-19-853248-4.

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