For any
n ×
n matrix
D ∈ ℝ
n×n
let adj
D denote the transpose of its cofactors. If det
D > 0 then there exists a symmetric matrix
$
\mathcal{A} = \mathcal{A}\left( \mathcal{D} \right)
$
\mathcal{A} = \mathcal{A}\left( \mathcal{D} \right)
with det
$
\mathcal{A}
$
\mathcal{A}
= 1 such that
|
$
adjD = \left( {detD} \right)^{\tfrac{{n - 2}}
{n}} \mathcal{A}D^l
$
adjD = \left( {detD} \right)^{\tfrac{{n - 2}}
{n}} \mathcal{A}D^l
|
Where
D’ is the transpose of
D.
For
n = 2,
K ≥ 1, the set of
K -quasiconformal matrices is denoted by
|
$
Q_2 \left( K \right) = \left\{ {D \in \mathbb{R}^{2 \times 2} :\left\| D \right\|^2 \leqslant K det D} \right\}
$
Q_2 \left( K \right) = \left\{ {D \in \mathbb{R}^{2 \times 2} :\left\| D \right\|^2 \leqslant K det D} \right\}
|
Furthermore define
|
$
\varepsilon _2 \left( K \right) = \left\{ {\mathcal{A} \in \mathbb{R}^{2 \times 2} :\frac{I}
{K} \leqslant \mathcal{A}^l = A \leqslant KI, det \mathcal{A} = 1} \right\}
$
\varepsilon _2 \left( K \right) = \left\{ {\mathcal{A} \in \mathbb{R}^{2 \times 2} :\frac{I}
{K} \leqslant \mathcal{A}^l = A \leqslant KI, det \mathcal{A} = 1} \right\}
|
For variable matrices
D =
D(
x)∈
ɛ
2(
K) for a.e.
x ∈ Ω ⊂ ℝ
2×2, Ω a simply connected and bounded domain, a natural question is to see how does
$
\mathcal{A} = \mathcal{A}\left( {x,\mathcal{D}} \right)
$
\mathcal{A} = \mathcal{A}\left( {x,\mathcal{D}} \right)
(
x,
D) change with
D(
x).