Criteria for finding extreme values in N-dimensions

4.8 Criteria for finding extreme values in N-dimensions

Definition 42 If \(\tilde A\) is a symmetric \(N\times N\) matrix, \(\vec x\in\mathbb{R}^N\) than:
\(\tilde A\) is positive definite if \(\vec x\cdot\tilde A\vec{x}\gt0\) for all \(\vec{x}\)
\(\tilde A\) is positive semi-definite if \(\vec x\cdot\tilde A\vec{x}\ge0\) for all \(\vec{x}\)
\(\tilde A\) is negative definite if \(\vec x\cdot\tilde A\vec{x}\lt0\) for all \(\vec{x}\)
\(\tilde A\) is negative semi-definite if \(\vec x\cdot\tilde A\vec{x}\le0\) for all \(\vec{x}\)
\(\tilde A\) is called indefinite if \(\vec x,\vec y\) exist with \(\vec x\cdot\tilde{A}\vec{x}\gt0\) and \(\vec y\cdot\tilde{A}\vec{y}\lt0\)

Criterion for extreme values in \(N\)-dimensions:
\(f:\mathbb{R}^N\to\mathbb{R}\;\;\vec{x}_0\in\mathbb{R}^N\) extreme value (local) than \(\vec\nabla f(\vec{x}_0)=0\)
if:
\(\tilde H(\vec{x}_0)\) positive definite\(\to\) minimum
\(\tilde H(\vec{x}_0)\) negative definite\(\to\) maximum
\(\tilde H(\vec{x}_0)\) indefinite\(\to\) no extremum (saddle point)
\(\tilde H(\vec{x}_0)\) positive semi-definite or neg. semi-definite then no decision is possible
Note: 1D is special case.
Hurwitz criterion: \(\tilde{A}\;\;N\times N\) matrix, symmetric

\[\tilde{A}=\left(\begin{array}{ccc}a_{11}&\cdots&a_{1N}\\\vdots&\ddots&\vdots\\a_{N1}&\cdots&a_{NN}\end{array}\right)\]

is positive definite if and only if

\[ \det \left|\begin{array}{ccc}a_{11}&\cdots&a_{1k}\\\vdots&\ddots&\vdots\\a_{k1}&\cdots&a_{kk}\end{array}\right|\gt0 \]

for all \(k=1,\ldots,N\) .

\(\tilde{A}\) is negative definite if \(- \tilde{A}\) is positive definite according to the Hurwitz criterion.
Otherwise \(\tilde{A}\) is indefinite, or semi-case.
Other criteria:
\(\tilde A\) symmetric, positive definite \(\Rightarrow\) all Eigenvalues positive
\(\tilde A\) symmetric, negative definite \(\Rightarrow\) all Eigenvalues negative
\(\tilde A\) symmetric semi cases\(\Rightarrow\) \(\lambda=0\) possible
otherwise indefinite

4.8.1 Examples for finding extrema