6-3 奇異??奇異??

@ӯx} $A$ P_ȡ]Singular Value^$\sigma$ Ω_Vq ]Singular Vectors^$u$ P $v$ sbUCYG $$ \left\{ \begin{matrix} Av=\sigma u\\ A^Tu=\sigma v \end{matrix} \right. $$ YNҦVq $u$ ñƦx} $U$AҦVq $v$ ñƦx} $V$AhWigG

$$ \left\{ \begin{matrix} AV=U\Sigma\\ A^TU=V\Sigma \end{matrix} \right. $$ 䤤 $\Sigma$ D﨤uYO $\sigma$ ȡAlsC

Y $A$ ׬O mnAh $U$B$U$B$V$ פOO mmBmn H nnC @ӨA$U$ M $V$ O Unitary x}]Yx}VqۤABVqק 1^AUCG

$$ \left\{ \begin{matrix} UU^T=I\\ VV^T=I \end{matrix} \right. $$ ]x} $A$ ig

$$A=U\Sigma V^T$$ W٬_Ȥѡ]Singular Value Decomposition^C

MATLAB svd OiΩpx}_ȤΩ_VqAڭ̨åiHPҩ_ȤѡApUG

Example 1: 06-uʥN/svd01.mA = [3 5; 4 7; 2 1; 0 3]; [U, S, V] = svd(A) maxDiff = max(max(abs(A-U*S*V')))U = -0.5577 0.0881 -0.6954 0.4447 -0.7714 0.0333 0.2489 -0.5848 -0.1771 0.6471 0.5453 0.5025 -0.2504 -0.7565 0.3965 0.4558 S = 10.4517 0 0 1.9397 0 0 0 0 V = -0.4892 0.8722 -0.8722 -0.4892 maxDiff = 3.5527e-015

Y $A$ mn x}B m >> nAhڭ̥ib svd O[Jt@ӿJ޼ 0AϨ䲣ͪ $U$ $S$ x}㦳pסAҦpG

Example 2: 06-uʥN/svd02.mA = [3 5; 4 7; 2 1; 0 3]; [U, S, V] = svd(A, 0) maxDiff = max(max(abs(A-U*S*V')))U = -0.5577 0.0881 -0.7714 0.0333 -0.1771 0.6471 -0.2504 -0.7565 S = 10.4517 0 0 1.9397 V = -0.4892 0.8722 -0.8722 -0.4892 maxDiff = 3.5527e-015

@ӨAY $A$ mn]m >> n^x}Ahؤ覡Ҳͪ $U$B $S$B $V$ x}AפOO mnBnn nnC

M svd O| svds gsvdAŪ̥iѽuW䴩dTC


MATLAB{]pGig