covariance matrix

probability&statics

Reference

geometric-interpretation-covariance-matrix/

结论性总结

• While the eigenvectors represent the rotation matrix, the eigenvalues correspond to the square of the scaling factor in each dimension.
• the largest eigenvector of the covariance matrix always points into the direction of the largest variance of the data, and the magnitude of this vector equals the corresponding eigenvalue. The second largest eigenvector is always orthogonal to the largest eigenvector, and points into the direction of the second largest spread of the data。

• eigenvalues represent the variance of the data along the eigenvector directions, whereas the variance components of the covariance matrix represent the spread along the axes. If there are no covariances, then both values are equal.

• 特征向量表示旋转矩阵，特征值表示缩放幅度的平方。

• 协方差矩阵的最大特征向量表示数据最大方差的方向；特征向量的幅度用特征值表示。
• 协方差矩阵的对角线位置放的是数据相同维度的方差（variance）；其他位置表示数据不同维度的协方差（covariance）。
• 特征值表示沿着特征向量方向数据的方差大小；协方差矩阵里面的方差元素(variance)表示沿着坐标轴方向的方差大小。

多看几遍再加深一下理解。