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Mathematics-Online lexicon: Annotation to

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:


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.


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  automatisch erstellt am 25.  6. 2018