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Diagonal Quadratic Form

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If A=(a_(ij)) is a diagonal matrix, then

Q(v)=v^(T)Av=suma_(ii)v_i^2
(1)

is a diagonal quadratic form, and Q(v,w)=v^(T)Aw is its associated diagonal symmetric bilinear form.

For a general symmetric matrix A, a symmetric bilinear form Q may be diagonalized by a nondegenerate n×n matrix C such that Q(Cv,Cw) is a diagonal form. That is, C^(T)AC is a diagonal matrix. Note that C may not be an orthogonal matrix.

For example, consider

A=[1 2; 2 3].
(2)

Then taking the diagonalizer

C=[1 -2; 0 1]
(3)

gives the diagonal matrix

C^(T)AC=[1 0; 0 -1].
(4)

See also

Diagonal Matrix, Matrix Signature, Quadratic Form, Symmetric Bilinear Form, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Diagonal Quadratic Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DiagonalQuadraticForm.html

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