# Sylvester's law of inertia

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Sylvester's law of inertia

In linear algebra, Sylvester's law of inertia is a theorem describing a canonical representative for a real symmetric matrix under congruence transformations. It is named for J. J. Sylvester who stated and proved it in 1852.

The theorem states that a real symmetric matrix is congruent to exactly one diagonal matrix with diagonal entries all being +1,-1 or zero.

The "inertia" is defined as the triple containing the numbers of diagonal entries which are +1, -1 and 0 respectively. These numbers are equal to the numbers of positive, negative and zero eigenvalues of "A": see also signature (quadratic form). A congruence transformation of "A" is formed as the product

:$SAS^T$

where "S" is a non-singular matrix. In other words, the signature of "A" as quadratic form is well-defined and independent under congruence transformations.

ee also

*Metric signature

References

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* [http://planetmath.org.sixxs.org/encyclopedia/SylvestersLaw.html Sylvester's law] on PlanetMath.

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