特征值与谱分解Eigenvalues & Spectral Decomposition

保持方向的向量Directions that stay on their line

若 Av=λv 且 v 非零，变换后仍在 v 所在直线上；λ 表示带符号的缩放。负特征值使箭头反向，零特征值把该方向压到原点。If Av=λv for nonzero v, the transformed vector stays on the line through v; λ is a signed scale factor. A negative eigenvalue reverses the arrow, and a zero eigenvalue collapses that direction to the origin.

对称矩阵有正交特征基Symmetric matrices have an orthogonal eigenbasis

实对称矩阵可以分解为 QΛQᵀ，其中 Q 的列互相垂直且长度为 1。重复特征值允许多个特征方向；不能把这个结论直接套给所有方阵。A real symmetric matrix can be decomposed as QΛQᵀ with mutually orthogonal unit columns in Q. A repeated eigenvalue permits multiple eigen-directions; this conclusion does not apply directly to every square matrix.

连接协方差主轴Connect to covariance axes

协方差矩阵是实对称且半正定的，其特征向量给出数据变化的正交主轴，特征值表示相应方向上的方差。一般矩阵甚至可能没有实特征方向，例如平面 90° 旋转。A covariance matrix is real symmetric and positive semidefinite. Its eigenvectors give orthogonal axes of variation and eigenvalues the corresponding variances. A general matrix may have no real eigen-direction, such as a 90° rotation in the plane.

公式与符号Formula and symbols

A 是方阵，v 是非零特征向量，λ 是其特征值；对于实对称矩阵，可用正交矩阵 Q 的列作为特征向量，Λ 的对角线是对应特征值。A is square, v a nonzero eigenvector, and λ its eigenvalue. For real symmetric A, the columns of orthogonal Q can be eigenvectors and the diagonal of Λ contains their eigenvalues.

继续探索Explore further

• 深入阅读：特征向量探针与 PCAFurther reading: eigenvector probe and PCA