Diagonal Quadratic Form
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
| [画像: A=[1 2; 2 3]. ] |
(2)
|
![画像: A=[1 2; 2 3].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/DiagonalQuadraticForm/NumberedEquation2.svg){kind=link}
Then taking the diagonalizer
| [画像: C=[1 -2; 0 1] ] |
(3)
|
![画像: C=[1 -2; 0 1]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/DiagonalQuadraticForm/NumberedEquation3.svg){kind=link}
gives the diagonal matrix
| [画像: C^(T)AC=[1 0; 0 -1]. ] |
(4)
|
![画像: C^(T)AC=[1 0; 0 -1].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/DiagonalQuadraticForm/NumberedEquation4.svg){kind=link}
See also
Diagonal Matrix, Matrix Signature, Quadratic Form, Symmetric Bilinear Form, Vector SpaceThis entry contributed by Todd Rowland
Explore with Wolfram|Alpha
WolframAlpha
More things to try:
Cite this as:
Rowland, Todd. "Diagonal Quadratic Form." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DiagonalQuadraticForm.html