矩阵的特征值和特征向量的含义请参考
引用知乎 :
特征值首先是描述特征的。比如你的图片是有特征的，并且图片是存在某个坐标系的。特征向量就代表这个坐标系，特征值就代表这个特征在这个坐标方向上的贡献。总之，就是代表在对应坐标轴上的特征大小的贡献.
在R中如何计算特征值和特征向量?
可以通过对矩阵A进行谱分解来得到矩阵的特征值和特征向量。矩阵A的谱分解如下：A=UΛU’，其中U的列为A的特征值
所对应的特征向量，在R中可以用eigen()函数得到U和Λ。例如：
eigen函数参数如下 :
其中，x参数输入矩阵；symmetric参数判断矩阵是否为对称矩阵，如果参数为空，系统将自动检测矩阵的对称性。例如：
eigen(A)得到一个list, 存储特征值和特征向量.
得到矩阵A的特征值：
得到矩阵A的特征向量：
> args(eigen) function (x, symmetric, only.values = FALSE, EISPACK = FALSE) NULLAI 代码解读
其中，x参数输入矩阵；symmetric参数判断矩阵是否为对称矩阵，如果参数为空，系统将自动检测矩阵的对称性。例如：
> A=matrix(1:9,nrow=3,ncol=3) > A [,1] [,2] [,3] [1,] 1 4 7 [2,] 2 5 8 [3,] 3 6 9AI 代码解读
eigen(A)得到一个list, 存储特征值和特征向量.
> class(eigen(A)) [1] "list" > Aeigen=eigen(A) > Aeigen $values [1] 1.611684e+01 -1.116844e+00 -4.054214e-16 $vectors [,1] [,2] [,3] [1,] -0.4645473 -0.8829060 0.4082483 [2,] -0.5707955 -0.2395204 -0.8164966 [3,] -0.6770438 0.4038651 0.4082483AI 代码解读
得到矩阵A的特征值：
> Aeigen$values [1] 1.611684e+01 -1.116844e+00 -4.054214e-16AI 代码解读
得到矩阵A的特征向量：
> Aeigen$vectors [,1] [,2] [,3] [1,] -0.4645473 -0.8829060 0.4082483 [2,] -0.5707955 -0.2395204 -0.8164966 [3,] -0.6770438 0.4038651 0.4082483AI 代码解读
[参考]
4. > help(eigen)
eigen package:base R Documentation Spectral Decomposition of a Matrix Description: Computes eigenvalues and eigenvectors of numeric (double, integer, logical) or complex matrices. Usage: eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE) Arguments: x: a numeric or complex matrix whose spectral decomposition is to be computed. Logical matrices are coerced to numeric. symmetric: if ‘TRUE’, the matrix is assumed to be symmetric (or Hermitian if complex) and only its lower triangle (diagonal included) is used. If ‘symmetric’ is not specified, the matrix is inspected for symmetry. only.values: if ‘TRUE’, only the eigenvalues are computed and returned, otherwise both eigenvalues and eigenvectors are returned. EISPACK: logical. Defunct and ignored. Details: If ‘symmetric’ is unspecified, the code attempts to determine if the matrix is symmetric up to plausible numerical inaccuracies. It is faster and surer to set the value yourself. Computing the eigenvectors is the slow part for large matrices. Computing the eigendecomposition of a matrix is subject to errors on a real-world computer: the definitive analysis is Wilkinson (1965). All you can hope for is a solution to a problem suitably close to ‘x’. So even though a real asymmetric ‘x’ may have an algebraic solution with repeated real eigenvalues, the computed solution may be of a similar matrix with complex conjugate pairs of eigenvalues. Value: The spectral decomposition of ‘x’ is returned as components of a list with components values: a vector containing the p eigenvalues of ‘x’, sorted in _decreasing_ order, according to ‘Mod(values)’ in the asymmetric case when they might be complex (even for real matrices). For real asymmetric matrices the vector will be complex only if complex conjugate pairs of eigenvalues are detected. vectors: either a p * p matrix whose columns contain the eigenvectors of ‘x’, or ‘NULL’ if ‘only.values’ is ‘TRUE’. The vectors are normalized to unit length. Recall that the eigenvectors are only defined up to a constant: even when the length is specified they are still only defined up to a scalar of modulus one (the sign for real matrices). If ‘r <- eigen(A)’, and ‘V <- r$vectors; lam <- r$values’, then A = V Lmbd V^(-1) (up to numerical fuzz), where Lmbd =‘diag(lam)’. Source: By default ‘eigen’ uses the LAPACK routines ‘DSYEVR’, ‘DGEEV’, ‘ZHEEV’ and ‘ZGEEV’ whereas LAPACK is from <URL: http://www.netlib.org/lapack> and its guide is listed in the references. References: Anderson. E. and ten others (1999) _LAPACK Users' Guide_. Third Edition. SIAM. Available on-line at <URL: http://www.netlib.org/lapack/lug/lapack_lug.html>. Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_. Wadsworth & Brooks/Cole. Springer-Verlag Lecture Notes in Computer Science *6*. Wilkinson, J. H. (1965) _The Algebraic Eigenvalue Problem._ Clarendon Press, Oxford. See Also: ‘svd’, a generalization of ‘eigen’; ‘qr’, and ‘chol’ for related decompositions. To compute the determinant of a matrix, the ‘qr’ decomposition is much more efficient: ‘det’. Examples: eigen(cbind(c(1,-1), c(-1,1))) eigen(cbind(c(1,-1), c(-1,1)), symmetric = FALSE) # same (different algorithm). eigen(cbind(1, c(1,-1)), only.values = TRUE) eigen(cbind(-1, 2:1)) # complex values eigen(print(cbind(c(0, 1i), c(-1i, 0)))) # Hermite ==> real Eigenvalues ## 3 x 3: eigen(cbind( 1, 3:1, 1:3)) eigen(cbind(-1, c(1:2,0), 0:2)) # complex valuesAI 代码解读
