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Diagonal Quadratic Form

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A diagonal quadratic form is a quadratic form whose matrix A=(a_(ij)) is a diagonal matrix. It has the form

Q(v)=v^TAv=suma_(ii)v_i^2.
(1)

Over a field of characteristic different from 2, Q(v,w)=v^TAw is its associated diagonal symmetric bilinear form.

Over such a field, for a general symmetric matrix A, a symmetric bilinear form Q may be diagonalized by an n×n nonsingular matrix C such that Q(Cv,Cw) is a diagonal form (Conrad, p. 27). That is, C^TAC 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^TAC=[1 0; 0 -1].
(4)

See also

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

Portions of this entry contributed by Todd Rowland

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References

Conrad, K. "Bilinear Forms." Undated notes, Theorem 4.7. https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf.

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Diagonal Quadratic Form

Cite this as:

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

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