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  • [学习笔记] CS131 Computer Vision: Foundations and Applications:Lecture 3 线性代数初步

    向量和矩阵

    什么是矩阵/向量?

    Vectors and matrix are just collections of ordered numbers that represent something: movements in space, scaling factors, pixel brightness, etc. We'll define some common uses and standard operations on them.

    向量:列向量/行向量

    用处:

    • Vectos can represented an offset in 2D or 3D space; points are just vectors from the origion
    • data(pixels, gradient at an image key point)can be treated as a vector

    矩阵:在python中图像被表示为像素亮度矩阵, grayscale(m*n), color(m*n*3)

    算术运算:

    • addition
    • scaling
    • norm: vector/matrix

    • inner prodcut/dot product of vectors
    • product of matrix
    • transpose
    • determinant

    通常任何满足以下四种性质的函数都可以作为范数:

    • 非负性
    • 正定性
    • 齐次性
    • 三角不等式

    特殊矩阵

    • 单位阵
    • 对角阵
    • 对称阵
    • 反对阵矩阵

    变换矩阵

    矩阵可以用来对向量进行变换

    • scaling
    • rotation

    齐次系统/齐次坐标:

    变换矩阵最右列被加到原有向量中

    这里有时候会用到前面看的仿射矩阵affine matrix 的知识(见参考资料):

    平移(translation):

    缩放(scaling)

    旋转(rotation)

     

    平移旋转缩放

    矩阵的逆

    如果A的逆存在,A是可逆的或者是非奇异的non-singular; 否则是不可逆或者是奇异的.

    伪逆(pseudoinverse):在计算大型矩阵逆的时候,会伴随这浮点数问题,而且不是每个矩阵都有逆

    np.linalg(A,B) to solve AX = B

    如果没有具体解, 返回最近的一个解

    如果有多个解,返回最小的那个解

    阵的阶 

    the rank of a transforamtion matrix tells you hwo many dimensions it transforms a vector to.

    满秩; m*m 矩阵,阶数为m

    阶数 < 5, 奇异矩阵,逆不存在

    非方阵没有逆

    特征值和特征向量

    what is eigenvector?

    An eigenvector x of a linear transformation A is a non-zero vector that when A is applied to it, does not change direction. And only scales the eigenvector by the scalar value lambda, called an eigenvalue.

    $$Ax = lambda x$$

    $Ax = (lambda I)x,  ightarrow (lambda I - A)x = 0$, $x$ is non-zero, thus,  $|(lambda I - A)| = 0$

    性质:

    • $tr(A) = sum_{i=1}^n lambda_i$
    • $|A| = prod_{i=1}^n lambda_i$
    • the rank of A is equal to the number of non-zero eigenvalues of A
    • the eigenvalues of a diagonal matrix $D = diag(d_1,...d_n)$ are just the diagonal entries $d_1, ..., d_n$

    分形理论(spectral theroy)

     

    对角化(diagonalization)

    如果n*n矩阵有n个线性独立的特征向量,则它是可对角化的

    如果n*n矩阵有n个不同的特征值,则它是可对角化的

    对应着不同特征值的特征向量是线性独立的

    所有的特征向量方程可以写为:

    $$AV = VD$$

    $V in R^{n*n}$  V的列是A的特征向量,D为对角矩阵,对应着值为A的特征值. 如果A可以写为:$A = VDV^{-1}$则A可对角化

    特征值特征向量和对称对阵

    对称矩阵的性质:$A^{-1} = A^T$, A所有的特征值都是实数,A所有的特征向量都是正交的. 

    Some applications of eigenvlues:  PageRank, Schrodinger's equation, PCA

    矩阵代数

    矩阵梯度:

    Hessian Matrix

     

    参考资料:

    https://docs.microsoft.com/en-us/dotnet/framework/winforms/advanced/how-to-rotate-reflect-and-skew-images

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  • 原文地址:https://www.cnblogs.com/vincentcheng/p/7992335.html
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