Lecture 6 Finding Orthogonal Bases

1.怎么求得U?

Gram Schemidt Orthogonalization
Singular Value Decomposition

有$X$怎么找到$U$?

施密特正交化

残差 $X^{\prime}_2=X_2-aU_1$ ,所以 $U_2=\frac {X^{\prime}_2} {\lVert X^{\prime}_2 \rVert^2_{2}}=\left[\begin{array}{c} 0 \\ b \end{array}\right]/b=\left[\begin{array}{c} 0 \\ 1 \end{array}\right]$

2. 斯密特正交化的几何示例

施密特正交化计算

$\begin{align} &X^{\prime}_2=X_2-Projection(X_2 \ onto \ U_1) \ , \text{best fit of } X_2 \text{ as weighted } U_1 。 \\ &\text{best fit} = U_1w = U_1U^{T}_1X_2=P_{U_1}(X_2) \end{align}$

示例，施密特正交化具体过程和想法。

$U_1=\left[\begin{array}{c} 1/\sqrt{2} \\ 1/\sqrt{2}\\ 0 \end{array}\right], X_2=\left[\begin{array}{c} 1 \\ 3\\ 0 \end{array}\right]$

那么，

$U_1U_1^TX_2=\left[\begin{array}{c} 1/2 & 1/2& 0\\ 1/2 & 1/2& 0\\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{c} 1\\ 3\\ 0 \end{array}\right]=\left[\begin{array}{c} 2\\ 2\\ 0 \end{array}\right]$ $X_2-X_2-U_1U_1^TX_2=\left[\begin{array}{c} 1\\ 3\\ 0 \end{array}\right]-\left[\begin{array}{c} 2\\ 2\\ 0 \end{array}\right]=\left[\begin{array}{c} -1\\ 1\\ 0 \end{array}\right]\tag{1}$ $U_2=\left[\begin{array}{c} -1\\ 1\\ 0 \end{array}\right]/ \sqrt{2}=\left[\begin{array}{c} -\sqrt{2}/2\\ \sqrt{2}/2\\ 0 \end{array}\right]$