日: 2014年4月26日

Eigenvalues and eigenvectors

Let A = (a_{jk}) be an n \times n matrix and X a column vector. The equation

AX = \lambda X \cdots (21)

where \lambda is a number can be written as

\displaystyle \left( \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\ \cdots \\ x_n \end{array} \right) = \lambda\left( \begin{array}{c} x_1 \\ x_2 \\ \cdots \\ x_n \end{array} \right) \cdots (22)

or

\displaystyle \left. \begin{array}{ccc} (a_{11} - \lambda)x_1 + a_{12}x_2 + \cdots + a_{1n}x_n & = & 0 \\ a_{21}x_1 + (a_{22} - \lambda)x_2 + \cdots + a_{2n}x_n & = & 0 \\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots & = & 0 \\ a_{n1}x_1 + a_{n2}x_2 + \cdots + (a_{nn} - \lambda)x_n & = & 0 \end{array}\right\} \cdots(23)

The equation (23) will have non-trivial solution if and only if

\displaystyle \left| \begin{array}{cccc} a_{11} - \lambda & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} - \lambda & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} - \lambda \end{array} \right| = 0 \cdots(24)

which is a polynomial equation of degree n in \lambda. The roots of this equation are called eigenvalues or characteristic values of the matrix A. Corresponding to each eigenvalue there will be a solution X \ne 0, i.e. a non-trivial solution, which is called an eigenvector or characteristic vector belonging to the eigenvalue. The equation (24) can also be written

\det(A - \lambda I) = 0 \cdots(25)

and the equation in \lambda is often called the characteristic equation.

固有値と固有ベクトル

　 A = (a_{jk}) を n \times n 行列とし X を列ベクトルとしましょう．以下の方程式について

AX = \lambda X \cdots (21)

ここで \lambda は数であり以下のように記述できます．

\displaystyle \left( \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\ \cdots \\ x_n \end{array} \right) = \lambda\left( \begin{array}{c} x_1 \\ x_2 \\ \cdots \\ x_n \end{array} \right) \cdots (22)

あるいは

\displaystyle \left. \begin{array}{ccc} (a_{11} - \lambda)x_1 + a_{12}x_2 + \cdots + a_{1n}x_n & = & 0 \\ a_{21}x_1 + (a_{22} - \lambda)x_2 + \cdots + a_{2n}x_n & = & 0 \\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots & = & 0 \\ a_{n1}x_1 + a_{n2}x_2 + \cdots + (a_{nn} - \lambda)x_n & = & 0 \end{array}\right\} \cdots(23)

　方程式 (23) は以下の場合にのみ非自明解が存在します．

\displaystyle \left| \begin{array}{cccc} a_{11} - \lambda & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} - \lambda & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} - \lambda \end{array} \right| = 0 \cdots(24)

これは \lambda における n 次多項式です．この方程式の根は行列 A の 固有値 または 特性値 と呼びます．各々の固有値に対応して X \ne 0 なる解，すなわち非自明解が存在し，それらを固有値に属する 固有ベクトル または 特性ベクトル と呼びます．方程式 (24) はまたこのようにも記述できます．

\det(A - \lambda I) = 0 \cdots(25)

また \lambda における方程式はしばしば 特性方程式 と呼びます．