Week1-3

作者: 忻恆 | 来源:发表于2020-03-18 21:49 被阅读0次

    1. substitude:替换

    2. a over b : \mathcal {\frac{a} {b}}

    3. determinant,det,行列式

    4. {\rm A}^{\rm-1} = {\frac{1} {ad-bc}}\begin{pmatrix} d & -b\\ -c & a  \end{pmatrix},需要自己计算一遍, {ad-bc}就是determinant。因此det为0时,无法除以  0,此时no inverse matrix

    5. orthogonal matrix:正交矩阵,{\rm Q}^{\rm -1} = {\rm Q}^{\rm T} {\rm Q} {\rm Q}^{\rm T} = {\rm I} {\rm Q} = {\rm Q}^{\rm T} {\rm Q}  

    Q times Q transpose is multiplying the row of Q against the row of Q, equal to the identity matrix. 

    So, if the rows are the same, then you get one. If the rows are different, then you get zero. 

    That's the definition of orthonormality, orthonormal vectors.

    So, the rows of Q are orthonormal vectors. and the columns of Q are also orthonormal vectors. 

    6. Orthogonal Matrix 的重要特征:

    \begin{align}\lVert{\rm Q}{\rm x}\lVert^2 &= \left( {\rm Q}{\rm x} \right) ^ {\rm T}\left( {\rm Q}{\rm x} \right)\\&=  {\rm x}  ^ {\rm T} {\rm Q}^ {\rm T}  {\rm Q}{\rm x} \\&=  {\rm x}  ^ {\rm T}{\rm x}\\&= \lVert{\rm x}\lVert^2\end{align}

    orthogonal matrix all preserves norms.

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