院解7 京大数学系H28 基礎II 6 行列値正則関数

$\tilde{a_{jk}}$で$A(z)$の$(j,k)$成分の余因子を表し$\epsilon(\sigma)$で置換$\sigma$の符号を表すとする.各$ a_{jk}(z)$の正則性から$\text{det}A(z)$,$A(z)$は正則であり
$\dfrac{d}{dz} \text{det} A(z)=\dfrac{d}{dz}\displaystyle\sum_{\sigma\in S_N} \epsilon(\sigma) a_{1\ \sigma(1) }(z)a_{2\ \sigma(2)}(z)\cdots a_{N\ \sigma(N)}(z)$
$=\displaystyle\sum_{k=1}^{N}\sum_{\sigma\in S_N}\epsilon(\sigma) a_{1\ \sigma(1)}(z)\cdots a^{\prime}_{k\ \sigma(k)}(z)\cdots a_{N\ \sigma(N)}(z)$
$=\displaystyle \sum_{k=1}^{N}\begin{vmatrix} a_{11}(z) & \cdots & a_{1N}(z)\\ \vdots & & \vdots \\ a^{\prime}_{k1}(z) & \cdots & a^{\prime}_{kN}(z)\\ \vdots & &\vdots \\ a_{N1}(z)& \cdots & a_{NN}(z) \end{vmatrix} $
$=\displaystyle\sum_{k=1}^N \sum_{i=1}^N a^\prime _{ki}(z)\tilde{a_{ki}(z)}=\text{tr}((a^\prime_{ij}(z))\tilde{A(z)})$
(ただし$\tilde{A(z)}$は$A(z)$の余因子行列)
$=\text{tr}(A^\prime (z)(\text{det}A(z))A(z)^{-1})=\text{det}A(z)\text{tr}(A(z)^{-1}\dfrac{d}{dz}A(z))$.
よって$\dfrac{(\text{det}A(z))^\prime}{\text{det}A(z)}=\text{tr}(A(z)^{-1}\dfrac{d}{dz}A(z))$.ここで$\text{det}A(z)$の正則性と正則関数の一般論から$\dfrac{1}{2\pi i}\displaystyle\int_{C_\epsilon}\dfrac{(\text{det}A(z))^\prime}{\text{det}A(z)}dz$の値は$\text{det}A(z)$の零点$z=0$の位数である.