(1)

行列\(A,B\)をそれぞれ\(m\times n,s\times t\)行列とする。
\begin{align*} A\otimes\left(B\otimes C\right) & =A\otimes\left(\begin{array}{ccc} b_{1,1}C & \cdots & b_{1,t}C\\ \vdots & \ddots & \vdots\\ b_{s,1}C & \cdots & b_{s,t}C \end{array}\right)\\ & =\left(\begin{array}{ccc} a_{1,1}\left(\begin{array}{ccc} b_{1,1}C & \cdots & b_{1,t}C\\ \vdots & \ddots & \vdots\\ b_{s,1}C & \cdots & b_{s,t}C \end{array}\right) & \cdots & a_{1,n}\left(\begin{array}{ccc} b_{1,1}C & \cdots & b_{1,t}C\\ \vdots & \ddots & \vdots\\ b_{s,1}C & \cdots & b_{s,t}C \end{array}\right)\\ \vdots & \ddots & \vdots\\ a_{m,1}\left(\begin{array}{ccc} b_{1,1}C & \cdots & b_{1,t}C\\ \vdots & \ddots & \vdots\\ b_{s,1}C & \cdots & b_{s,t}C \end{array}\right) & \cdots & a_{m,n}\left(\begin{array}{ccc} b_{1,1}C & \cdots & b_{1,t}C\\ \vdots & \ddots & \vdots\\ b_{s,1}C & \cdots & b_{s,t}C \end{array}\right) \end{array}\right)\\ & =\left(\begin{array}{ccc} a_{1,1}BC & \cdots & a_{1,n}BC\\ \vdots & \ddots & \vdots\\ a_{m,1}BC & \cdots & a_{m,n}BC \end{array}\right)\\ & =\left(\begin{array}{ccc} a_{1,1}B & \cdots & a_{1,n}B\\ \vdots & \ddots & \vdots\\ a_{m,1}B & \cdots & a_{m,n}B \end{array}\right)\otimes C\\ & =\left(\left(\begin{array}{ccc} a_{1,1} & \cdots & a_{1,n}\\ \vdots & \ddots & \vdots\\ a_{m,1} & \cdots & a_{m,n} \end{array}\right)\otimes B\right)\otimes C\\ & =\left(A\otimes B\right)\otimes C \end{align*}

(2)

行列\(A\)を\(m\times n\)行列とする。
\begin{align*} A\otimes\left(B+C\right) & =\left(\begin{array}{cccc} a_{1,1}\left(B+C\right) & a_{1,2}\left(B+C\right) & \cdots & a_{1,n}\left(B+C\right)\\ a_{2,1}\left(B+C\right) & a_{2,2}\left(B+C\right) & \cdots & a_{2,n}\left(B+C\right)\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}\left(B+C\right) & a_{m.2}\left(B+C\right) & \cdots & a_{m,n}\left(B+C\right) \end{array}\right)\\ & =\left(\begin{array}{cccc} a_{1,1}B & a_{1,2}B & \cdots & a_{1,n}B\\ a_{2,1}B & a_{2,2}B & \cdots & a_{2,n}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}B & a_{m.2}B & \cdots & a_{m,n}B \end{array}\right)+\left(\begin{array}{cccc} a_{1,1}C & a_{1,2}C & \cdots & a_{1,n}C\\ a_{2,1}C & a_{2,2}C & \cdots & a_{2,n}C\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}C & a_{m.2}C & \cdots & a_{m,n}C \end{array}\right)\\ & =\left(\begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,n}\\ a_{2,1} & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1} & a_{m.2} & \cdots & a_{m,n} \end{array}\right)\otimes B+\left(\begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,n}\\ a_{2,1} & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1} & a_{m.2} & \cdots & a_{m,n} \end{array}\right)\otimes C\\ & =A\otimes B+A\otimes C \end{align*}

(3)

行列\(A,B\)を\(m\times n\)行列とする。
\begin{align*} \left(A+B\right)\otimes C & =\left(\begin{array}{cccc} \left(a_{1,1}+b_{1,1}\right)C & \left(a_{1,2}+b_{1,2}\right)C & \cdots & \left(a_{1,n}+b_{1,n}\right)C\\ \left(a_{2,1}+b_{2,1}\right)C & \left(a_{2,2}+b_{2,2}\right)C & \cdots & \left(a_{2,n}+b_{2,n}\right)C\\ \vdots & \vdots & \ddots & \vdots\\ \left(a_{m,1}+b_{m,1}\right)C & \left(a_{m.2}+b_{m,2}\right)C & \cdots & \left(a_{m,n}+b_{m,n}\right)C \end{array}\right)\\ & =\left(\begin{array}{cccc} a_{1,1}C & a_{1,2}C & \cdots & a_{1,n}C\\ a_{2,1}C & a_{2,2}C & \cdots & a_{2,n}C\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}C & a_{m.2}C & \cdots & a_{m,n}C \end{array}\right)+\left(\begin{array}{cccc} b_{1,1}C & b_{1,2}C & \cdots & b_{1,n}C\\ b_{2,1}C & b_{2,2}C & \cdots & b_{2,n}C\\ \vdots & \vdots & \ddots & \vdots\\ b_{m,1}C & b_{m.2}C & \cdots & b_{m,n}C \end{array}\right)\\ & =\left(\begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,n}\\ a_{2,1} & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1} & a_{m.2} & \cdots & a_{m,n} \end{array}\right)\otimes C+\left(\begin{array}{cccc} b_{1,1} & b_{1,2} & \cdots & b_{1,n}\\ b_{2,1} & b_{2,2} & \cdots & b_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ b_{m,1} & b_{m.2} & \cdots & b_{m,n} \end{array}\right)\otimes C\\ & =A\otimes C+B\otimes C \end{align*}

(4)

\begin{align*} k\left(A\otimes B\right) & =k\left(\begin{array}{cccc} a_{1,1}B & a_{1,2}B & \cdots & a_{1,n}B\\ a_{2,1}B & a_{2,2}B & \cdots & a_{2,n}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}B & a_{m.2}B & \cdots & a_{m,n}B \end{array}\right)\\ & =\left(\begin{array}{cccc} ka_{1,1}B & ka_{1,2}B & \cdots & ka_{1,n}B\\ ka_{2,1}B & ka_{2,2}B & \cdots & ka_{2,n}B\\ \vdots & \vdots & \ddots & \vdots\\ ka_{m,1}B & ka_{m.2}B & \cdots & ka_{m,n}B \end{array}\right)\\ & =\left(\begin{array}{cccc} ka_{1,1} & ka_{1,2} & \cdots & ka_{1,n}\\ ka_{2,1} & ka_{2,2} & \cdots & ka_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ ka_{m,1} & ka_{m.2} & \cdots & ka_{m,n} \end{array}\right)\otimes B\\ & =\left(k\left(\begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,n}\\ a_{2,1} & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1} & a_{m.2} & \cdots & a_{m,n} \end{array}\right)\right)\otimes B\\ & =\left(kA\right)\otimes B \end{align*} \begin{align*} k\left(A\otimes B\right) & =k\left(\begin{array}{cccc} a_{1,1}B & a_{1,2}B & \cdots & a_{1,n}B\\ a_{2,1}B & a_{2,2}B & \cdots & a_{2,n}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}B & a_{m.2}B & \cdots & a_{m,n}B \end{array}\right)\\ & =\left(\begin{array}{cccc} a_{1,1}kB & a_{1,2}kB & \cdots & a_{1,n}kB\\ a_{2,1}kB & a_{2,2}kB & \cdots & a_{2,n}kB\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}kB & a_{m.2}kB & \cdots & a_{m,n}kB \end{array}\right)\\ & =\left(\begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,n}\\ a_{2,1} & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1} & a_{m.2} & \cdots & a_{m,n} \end{array}\right)\otimes kB\\ & =A\otimes\left(kB\right) \end{align*}

(5)

行列\(A,B,C,D\)をそれぞれ\(l\times m,r\times s,m\times n,s\times t\)行列とする。
\begin{align*} \left(A\otimes B\right)\left(C\otimes D\right) & =\left(\begin{array}{cccc} a_{1,1}B & a_{1,2}B & \cdots & a_{1,m}B\\ a_{2,1}B & a_{2,2}B & \cdots & a_{2,m}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{l,1}B & a_{l.2}B & \cdots & a_{l,m}B \end{array}\right)\left(\begin{array}{cccc} c_{1,1}D & c_{1,2}D & \cdots & c_{1,n}D\\ c_{2,1}D & c_{2,2}D & \cdots & c_{2,n}D\\ \vdots & \vdots & \ddots & \vdots\\ c_{m,1}D & c_{m.2}D & \cdots & c_{m,n}D \end{array}\right)\\ & =\left(\begin{array}{cccc} \sum_{i=1}^{m}a_{1,i}c_{i,1}BD & \sum_{i=1}^{m}a_{1,i}c_{i,2}BD & \cdots & \sum_{i=1}^{m}a_{1,i}c_{i,m}BD\\ \sum_{i=1}^{m}a_{2,i}c_{i,1}BD & \sum_{i=2}^{m}a_{1,i}c_{i,2}BD & \cdots & \sum_{i=1}^{m}a_{2,i}c_{i,m}BD\\ \vdots & \vdots & \ddots & \vdots\\ \sum_{i=1}^{m}a_{l,i}c_{i,1}BD & \sum_{i=1}^{m}a_{1,i}c_{i,2}BD & \cdots & \sum_{i=1}^{m}a_{l,i}c_{i,m}BD \end{array}\right)\\ & =\left(\begin{array}{cccc} \sum_{i=1}^{m}a_{1,i}c_{i,1} & \sum_{i=1}^{m}a_{1,i}c_{i,2} & \cdots & \sum_{i=1}^{m}a_{1,i}c_{i,m}\\ \sum_{i=1}^{m}a_{2,i}c_{i,1} & \sum_{i=2}^{m}a_{1,i}c_{i,2} & \cdots & \sum_{i=1}^{m}a_{2,i}c_{i,m}\\ \vdots & \vdots & \ddots & \vdots\\ \sum_{i=1}^{m}a_{l,i}c_{i,1} & \sum_{i=1}^{m}a_{1,i}c_{i,2} & \cdots & \sum_{i=1}^{m}a_{l,i}c_{i,m} \end{array}\right)\otimes BD\\ & =\left(AC\right)\otimes\left(BD\right) \end{align*}

(6)

\begin{align*} \left(A\otimes B\right)\left(A^{-1}\otimes B^{-1}\right) & =\left(AA^{-1}\right)\otimes\left(BB^{-1}\right)\cmt{\because\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes\left(BD\right)}\\ & =I\otimes I\\ & =I \end{align*} となるので、
\[ \left(A\otimes B\right)^{-1}=A^{-1}\otimes B^{-1} \] が成り立つ。

(7)

\begin{align*} \overline{A\otimes B} & =\overline{\left(\begin{array}{cccc} a_{1,1}B & a_{1,2}B & \cdots & a_{1,n}B\\ a_{2,1}B & a_{2,2}B & \cdots & a_{2,n}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}B & a_{m.2}B & \cdots & a_{m,n}B \end{array}\right)}\\ & =\left(\begin{array}{cccc} \overline{a_{1,1}B} & \overline{a_{1,2}B} & \cdots & \overline{a_{1,n}B}\\ \overline{a_{2,1}B} & \overline{a_{2,2}B} & \cdots & \overline{a_{2,n}B}\\ \vdots & \vdots & \ddots & \vdots\\ \overline{a_{m,1}B} & \overline{a_{m.2}B} & \cdots & \overline{a_{m,n}B} \end{array}\right)\\ & =\left(\begin{array}{cccc} \overline{a_{1,1}}\overline{B} & \overline{a_{1,2}}\overline{B} & \cdots & \overline{a_{1,n}}\overline{B}\\ \overline{a_{2,1}}\overline{B} & \overline{a_{2,2}}\overline{B} & \cdots & \overline{a_{2,n}}\overline{B}\\ \vdots & \vdots & \ddots & \vdots\\ \overline{a_{m,1}}\overline{B} & \overline{a_{m.2}}\overline{B} & \cdots & \overline{a_{m,n}}\overline{B} \end{array}\right)\\ & =\overline{A}\otimes\overline{B} \end{align*}

(8)

\begin{align*} \left(A\otimes B\right)^{T} & =\left(\begin{array}{cccc} a_{1,1}B & a_{1,2}B & \cdots & a_{1,n}B\\ a_{2,1}B & a_{2,2}B & \cdots & a_{2,n}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1}B & a_{m.2}B & \cdots & a_{m,n}B \end{array}\right)^{T}\\ & =\left(\begin{array}{cccc} a_{1,1}B^{T} & a_{2,1}B^{T} & \cdots & a_{m,1}B^{T}\\ a_{1,2}B^{T} & a_{2,2}B^{T} & \cdots & a_{m,2}B^{T}\\ \vdots & \vdots & \ddots & \vdots\\ a_{1,n}B^{T} & a_{2,n}B^{T} & \cdots & a_{m,n}B^{T} \end{array}\right)\\ & =\left(\begin{array}{cccc} a_{1,1} & a_{2,1} & \cdots & a_{m,1}\\ a_{1,2} & a_{2,2} & \cdots & a_{m,2}\\ \vdots & \vdots & \ddots & \vdots\\ a_{1,n} & a_{2,n} & \cdots & a_{m,n} \end{array}\right)\otimes B^{T}\\ & =A^{T}\otimes B^{T} \end{align*}

(9)

\begin{align*} \left(A\otimes B\right)^{*} & =\overline{\left(A\otimes B\right)^{T}}\\ & =\overline{A^{T}\otimes B^{T}}\\ & =\overline{A^{T}}\otimes\overline{B^{T}}\\ & =A^{*}\otimes B^{*} \end{align*}

(10)

\(A\)の固有値\(\lambda_{1},\lambda_{2},\cdots,\lambda_{m}\)と固有ベクトル\(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\cdots,\boldsymbol{x}_{m}\)は\(i\in\left\{ 1,2,\cdots,m\right\} ,A\boldsymbol{x}_{i}=\lambda_{i}\boldsymbol{x}\)となる。
同様に\(B\)の固有値\(\mu_{1},\mu_{2},\cdots,\mu_{n}\)と固有ベクトル\(\boldsymbol{y}_{1},\boldsymbol{y}_{2},\cdots,\boldsymbol{y}_{n}\)は\(i\in\left\{ 1,2,\cdots,n\right\} ,B\boldsymbol{y}_{i}=\mu_{i}\boldsymbol{y}\)となる。
これより、
\begin{align*} \left(A\otimes B\right)\left(\boldsymbol{x}_{i}\otimes\boldsymbol{y}_{j}\right) & =A\boldsymbol{x}_{i}\otimes B\boldsymbol{y}_{j}\\ & =\lambda_{i}\boldsymbol{x}_{i}\otimes\mu_{j}\boldsymbol{y}_{j}\\ & =\lambda_{i}\boldsymbol{x}_{j}\otimes\boldsymbol{y}_{j} \end{align*} となるので題意は成り立つ。

(11)

行列のトレースは重複も入れた全ての固有値の和なので、
\begin{align*} \tr\left(A\otimes B\right) & =\sum_{i=1}^{m}\sum_{j=1}^{n}\lambda_{i}\mu_{j}\\ & =\sum_{i=1}^{m}\lambda_{i}\sum_{j=1}^{n}\mu_{j}\\ & =\tr\left(A\right)\tr\left(B\right) \end{align*} となり、与式は成り立つ。

(12)

行列の行列式は重複も入れた全ての固有値の積なので、
\begin{align*} \det\left(A\otimes B\right) & =\prod_{i=1}^{m}\prod_{j=1}^{n}\lambda_{i}\mu_{j}\\ & =\prod_{i=1}^{m}\lambda_{i}^{n}\prod_{j=1}^{n}\mu_{j}\\ & =\left(\prod_{i=1}^{m}\lambda_{i}^{n}\right)\left(\prod_{j=1}^{n}\mu_{j}^{m}\right)\\ & =\left(\det A\right)^{n}\left(\det B\right)^{m} \end{align*} となり、与式は成り立つ。