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,
![画像: A=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 -1]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/MatrixSignature/NumberedEquation1.svg){kind=link}
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 SpaceThis entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd. "Matrix Signature." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MatrixSignature.html