## 1. 正交的一些概念和性质

1. $Q^TQ = I$

2. 若 P 和 Q 是标准正交矩阵，那么 X = PQ 也是标准正交矩阵。

3. 正交矩阵最重要的性质之一是它的变换可以保证一个向量的长度不变，包括 Euclidean lenght , matrix norm 和 Frobenius norm.

## 2. 正交矩阵

$e^{i(-\theta)} = cos(-\theta) + \mathb{i}sin(-\theta) = cos\theta - \mathb{i}sin\theta$

$(cos\theta - \mathb{i}sin\theta)(a + \mathb{i}b) = (acos\theta + bsin\theta) + \mathb{i}(a(-sin\theta) + bcos\theta)$

## 3. Householder Transformations

1. ,
2. ,

The goal is to find a linear transformation that changes the vector $x$ into a vector of same length which is collinear to $e_1$. We could use an orthogonal projection (Gram-Schmidt) but this will be numerically unstable if the vectors $x$ and $e_1$ are close to orthogonal. Instead, the Householder reflection reflects through the dotted line (chosen to bisect the angle between $x$ and $e_1$). The maximum angle with this transform is at most 45 degrees.

posted on 2013-07-22 15:10  daniel-D  阅读(4125)  评论(0编辑  收藏  举报