LinearSystemofComplexRoots

例題次の微分方程式の基本解を求めよ．

$\displaystyle {\bf X}^{\prime} = \left(\begin{array}{rrr} 2&-1&-1\\ 2&1&-1\\ 0&-1&1 \end{array}\right){\bf X}$

解

$\displaystyle \det(A - \lambda I) = \det\left(\begin{array}{rrr} 2-\lambda&-1&-... ...&-1&1-\lambda \end{array}\right) = (2-\lambda)(\lambda^{2} - 2\lambda + 2) = 0$

より固有値 $\lambda = 2, 1 \pm i$ を得る．

まず，固有値 $2$ に対応する固有ベクトルを求めると

$\displaystyle A - 2I$	$\displaystyle =$	$\displaystyle \left(\begin{array}{rrr} 0&-1&-1\\ 2&-1&-1\\ 0&-1&-1 \end{array... ...rrow} \left(\begin{array}{rrr} 1&-1/2&-1/2\\ 0&1&1\\ 0&0&0 \end{array}\right)$
	$\displaystyle \stackrel{\frac{R_{2}}{2}+ R_{1}\to R_{1}} {\longrightarrow}$	$\displaystyle \left(\begin{array}{rrr} 1&0&0\\ 0&1&1\\ 0&0&0 \end{array}\right)$

より固有ベクトルは ${\bf C} = \left(\begin{array}{r} 0\\ -1\\ 1 \end{array}\right)$ である．

次に固有値 $1+i$ に対応する固有ベクトルを求める．

$\displaystyle A - (1+i) I$	$\displaystyle =$	$\displaystyle \left(\begin{array}{rrr} 1-i&-1&-1\\ 2&-i&-1\\ 0&-1&-i \end{array}\right)$
	$\displaystyle \stackrel{\begin{array}{c} {}_{\frac{1}{1-i} \times R_{1}\to R_{1}}\\ {}_{-(1+i)R_{1} + R_{2}\to R_{2}} \end{array}}{\longrightarrow}$	$\displaystyle \left(\begin{array}{rrr} 1&-1/1-i&-1/1-i\\ 0&1&i\\ 0&-1&-i \end{array}\right)$
	$\displaystyle \stackrel{\begin{array}{c} {}_{\frac{R_{2}}{1-i} + R_{1}\to R_{1}}\\ {}_{R_{2} + R_{3}\to R_{3}} \end{array}}{\longrightarrow}$	$\displaystyle \left(\begin{array}{rrr} 1&0&-1\\ 0&1&i\\ 0&0&0 \end{array}\right)$

より固有ベクトルは ${\bf C} = \left(\begin{array}{r} 1\\ -i\\ 1 \end{array}\right)$ である．これより ${\bf C}e^{\lambda t}$ の実部と虚部を求めると，

$\displaystyle {\bf C}e^{\lambda t}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c} 1\\ -i\\ 1 \end{array}\right)e^{(1+i)t} = \left(\begin{array}{c} 1\\ -i\\ 1 \end{array}\right)e^{t}(\cos{t} + i\sin{t})$
	$\displaystyle =$	$\displaystyle \left(\begin{array}{c} e^{t}\cos{t} + ie^{t}\sin{t}\\ e^{t}\sin{... ...}\\ - e^{t}\cos{t}\\ e^{t}\sin{t} \end{array}\right)}_{\mbox{imaginary part}}$

となる．これより．基本解は

$\displaystyle \Phi = \begin{pmatrix}0& e^{t}\cos{t} & e^{t}\sin{t}\\ e^{2t} & ... ...}\sin{t} & -e^{t}\cos{t}\\ -e^{2t} & e^{t}\cos{t} & e^{t}\sin{t} \end{pmatrix}$

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