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 a symmetric matrix, then for any invertible matrix $S$ , the number of positive, negative and zero eigenvalues (called the inertia of the matrix) of $D=SAS^{\mathrm {T} }$ is constant. This result is particularly useful when $D$ is diagonal, as the inertia of a diagonal matrix can easily be obtained by looking at the sign of its diagonal 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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