6-2 ????????

@Ӥ} A TVq]Eigenvector^$x$ PTȡ]Eigenvalue^$\lambda$ YpUG $$ Ax = \lambda x$$ Wi² $$(A-\lambda I)x=0$$ ѩ $x$ O@ӹsVqA] $A-\lambda I$ O SingularAW~|ѡC $A-\lambda I$ O Singular ɡACsG $$|A-\lambda I|=0$$ Y A nn x}AhW $\lambda$ n h AN $\lambda$ N n Ӹ $\lambda_1, \lambda_2, \dots, \lambda_n$AUCYG $$ \left\{ \begin{matrix} A x_1 = \lambda_1 x_1 \\ \vdots\\ A x_2 = \lambda_2 x_2 \end{matrix} \right. $$ Υigx}ΦG $$AX=XD$$ 䤤

$$X= \begin{bmatrix} | & & | \\ x_1 & \dots & x_n \\ | & & | \end{bmatrix}, D= \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n \end{bmatrix} $$ pG $X^{-1}$ sbAhѤWio $$A=XDX^{-1}$$ ٬TȤѡ]Eigenvalue Decomposition^C

MATLAB eig OiΩpx}TȤΩTVqCYuQpTȮɡAiJpUG

Example 1: 06-uʥN/eig01.mA = magic(5); lambda = eig(A)lambda = 65.0000 -21.2768 -13.1263 21.2768 13.1263

YnPɨoTȤΩTVqɡAѨӿX޼ơAPɧڭ̤]iHҩTȤѡApUG

Example 2: 06-uʥN/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

䤤 X C@欰@өTVqA D 﨤uhOTȡC

Hint
  • TVquwVAӤwסA]Τ@_AMATLAB |i楿WơAN X C@ӦVq׳]w 1C

Hint
  • Y inv(X) sbAhN A TȤѦsbAY A iܦ XDX1䤤 D u﨤u^Ax} A ٬ Diagonalizable Not DefectiveC
  • Y A O DiagonalizableAhڭ̥iNܦ Jordan Canonical FormAiϥ Symbolic Math Toolbox jordan Oӭp⤧C


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