Matrix Signature

A real, nondegenerate n×n symmetric matrix A, and its corresponding symmetric bilinear form Q(v,w)=v^(T)Aw, has signature (p,q) if there is a nondegenerate matrix C such that C^(T)AC is a diagonal matrix with p 1s and q -1s. In this case, Q(Cv,Cw) is a diagonal quadratic form.

For example,

 A=[1 0 0  0; 0 1 0  0; 0 0 1  0; 0 0 0 -1]

gives a symmetric bilinear form Q called the Lorentzian inner product, which has signature (3,1).

See also

Diagonal Quadratic Form, Orthogonal Group, Quadratic Form, Symmetric Bilinear Form, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Matrix Signature." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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