野村数学研究所

論理+数式

行列

連立1次方程式と拡大係数行列の定義と性質

\[ \left(A,\boldsymbol{b}\right)=\left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_{1}\\ a_{21} & a_{22} & \cdots & a_{2n} & b_{2}\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_{m} \end{array}\right) \]

行列

(*)階数の性質

\[ \rank\left(AB\right)\leq\min\left(\rank\left(A\right),\rank\left(B\right)\right) \]

行列

クラメルの公式

\[ x_{i}=\frac{\det\left(\boldsymbol{a}_{1},\boldsymbol{a}_{2},\cdots,\boldsymbol{a}_{i-1},\boldsymbol{b},\boldsymbol{a}_{i+1},\cdots\boldsymbol{a}_{n}\right)}{\det\left(A\right)} \]

行列

行基本変形・列基本変形と基本行列

\[ \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{array}\right)\left(\begin{array}{cccc} a & b & c & d\\ e & f & g & h\\ i & j & k & l \end{array}\right)=\left(\begin{array}{cccc} a & b & c & d\\ i & j & k & l\\ e & f & g & h \end{array}\right) \]

行列

行列式の基本性質

\[ \det\left(\boldsymbol{a}_{\sigma\left(1\right)},\boldsymbol{a}_{\sigma\left(2\right)},\cdots,\boldsymbol{a}_{\sigma\left(n\right)}\right)=\sgn\left(\sigma\right)\det\left(\boldsymbol{a}_{1},\boldsymbol{a}_{2},\cdots,\boldsymbol{a}_{n}\right) \]

多重対数関数

逆数の多重対数関数

\[ \Li_{n}\left(\frac{1}{z}\right)=\left(-1\right)^{n+1}\Li_{n}\left(z\right)+\left(1+\left(-1\right)^{n}\right)\zeta\left(n\right)+\left(-1\right)^{n}\sum_{k=0}^{\left\lfloor \frac{n-3}{2}\right\rfloor }\left\{ 2\zeta\left(2\left(k+1\right)\right)\frac{\Log^{n-2\left(k+1\right)}z}{\left(n-2\left(k+1\right)\right)!}\right\} +\left(-1\right)^{n+1}\frac{\Log^{n}z}{n!}+\left(-1\right)^{n+1}\frac{\Log^{n-1}z}{\left(n-1\right)!}\left(\Log\left(1-z\right)-\Log\left(z-1\right)\right) \]