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

Quadratic Form

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For a (real) symmetric matrix $ A$ the expression

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

is called quadratic form.

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

(Authors: App/Burkhardt/Höllig)
It is not necessary to require matrix $ A$ 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 $ f$:

$\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 $ f$ has continuous second partial derivatives this matrix is symmetric.

(Authors: App/Burkhardt/Höllig)
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 $ 2$ 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$.

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005