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¤@­Ó¤è°} A ªº©T¦³¦V¶q¡]Eigenvector¡^$x$ »P©T¦³­È¡]Eigenvalue¡^$\lambda$ ªºÃö«Y¦¡¦p¤U¡G $$Ax = \lambda x$$ ¤W¦¡¥i¤ÆÂ²¦¨ $$(A-\lambda I)x=0$$ ¥Ñ©ó $x$ ¤£¬O¤@­Ó¹s¦V¶q¡A¦]¦¹ $A-\lambda I$ ¥²¶·¬O Singular¡A¤W¦¡¤~·|¦³¸Ñ¡C·í $A-\lambda I$ ¬O Singular ®É¡A¨ä¦æ¦C¦¡¬°¹s¡G $$|A-\lambda I|=0$$ ­Y A ¬° n¡Ñn ªº¯x°}¡A«h¤W¦¡¬° $\lambda$ ªº n ¦¸¦h¶µ¦¡ ¡A¥Nªí $\lambda$ ±N¦³ n ­Ó¸Ñ $\lambda_1, \lambda_2, \dots, \lambda_n$¡Aº¡¨¬¤U¦CÃö«Y¦¡¡G $$\left\{ \begin{matrix} A x_1 = \lambda_1 x_1 \\ \vdots\\ A x_2 = \lambda_2 x_2 \end{matrix} \right.$$ ©Î¥i¼g¦¨¯x°}§Î¦¡¡G $$AX=XD$$ ¨ä¤¤

$$X= \begin{bmatrix} | & & | \\ x_1 & \dots & x_n \\ | & & | \end{bmatrix}, D= \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n \end{bmatrix}$$ ¦pªG $X^{-1}$ ¦s¦b¡A«h¥Ñ¤W¦¡¥i±o $$A=XDX^{-1}$$ ¦¹¦¡ºÙ¬°©T¦³­È¤À¸Ñ¡]Eigenvalue Decomposition¡^¡C

Example 1: 06-½u©Ê¥N¼Æ/eig01.mA = magic(5); lambda = eig(A)lambda = 65.0000 -21.2768 -13.1263 21.2768 13.1263

Example 2: 06-½u©Ê¥N¼Æ/eig02.mA = magic(5); [X, D] = eig(A) maxDiff=max(max(abs(A-X*D*inv(X))))X = -0.4472 0.0976 -0.6330 0.6780 -0.2619 -0.4472 0.3525 0.5895 0.3223 -0.1732 -0.4472 0.5501 -0.3915 -0.5501 0.3915 -0.4472 -0.3223 0.1732 -0.3525 -0.5895 -0.4472 -0.6780 0.2619 -0.0976 0.6330 D = 65.0000 0 0 0 0 0 -21.2768 0 0 0 0 0 -13.1263 0 0 0 0 0 21.2768 0 0 0 0 0 13.1263 maxDiff = 2.8422e-014