Mathematics-Online course: Linear Algebra-Analytic Geometry-Quadrics-Quadratic Form

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	Mathematics-Online course: Linear Algebra - Analytic Geometry - Quadrics
	Quadratic Form

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For a (real) symmetric matrix

the expression

$\displaystyle a(x) =x^{\operatorname t}A x$

is called quadratic form.

Dependent on the sign of the eigenvalues of we distinguish between three types:

elliptic: All eigenvalues have the same sign.
parabolic: At least one eigenvalue equals zero and the other eigenvalues have the same sign.
hyperbolic: There are two eigenvalues of different sign.

It is not necessary to require matrix

to be symmetric. For an arbitrary matrix the quadratic form can be written as

$\displaystyle x^{\operatorname t}A x = \sum_{j,k} x_j a_{j,k} x_k = \sum_{j} a_{j,j} x_j^2 + \sum_{j<k} (a_{j,k}+a_{k,j}) x_jx_k \,.$

Hence, it is possible to symmetrise the quadratic form:

$\displaystyle x^{\operatorname t}A x = \frac{1}{2} x^{\operatorname t}(A + A^{\operatorname t}) x \,.$

Quadratic forms occur, for example, in the Taylor expansion of a scalar function :

$\displaystyle f(x) \rightarrow f(0) + \operatorname{grad}f(0)^{\operatorname t}\,x + \frac{1}{2} x^{\operatorname t}(\operatorname{H}f(0))\,x + \cdots \,.$

In this case $\operatorname{H}f$ is the Hesse matrix of the second partial derivatives. If

has continuous second partial derivatives this matrix is symmetric.

The symmetric matrix

$\begin{displaymath} A=\left( \begin{array}{cc} 1+\lambda&1-\lambda\\ 1-\lambda&1+\lambda \end{array}\right) \end{displaymath}$

has the eigenvalues

and $2\lambda$ .

Hence, the quadratic form

$\displaystyle a(x)=x^{\operatorname t}A x$

is elliptic for $\lambda > 0$ , parabolic for $\lambda=0$ and hyperbolic for $\lambda <0$ .

automatically generated 4/21/2005