numpy求矩阵的特征值与特征向量(np.linalg.eig函数用法)

Codefmeister 2023-03-20 我要评论

求矩阵的特征值与特征向量(np.linalg.eig)

语法

np.linalg.eig(a)

功能

Compute the eigenvalues and right eigenvectors of a square array.

求方阵(n x n)的特征值与右特征向量

Parameters

a : (…, M, M) array

Matrices for which the eigenvalues and right eigenvectors will be computed

a是一个矩阵Matrix的数组。每个矩阵M都会被计算其特征值与特征向量。

Returns

w : (…, M) array

The eigenvalues, each repeated according to its multiplicity.
The eigenvalues are not necessarily ordered. The resulting array will be of complex type, unless the imaginary part is zero in which case it will be cast to a real type. When a is real the resulting eigenvalues will be real (0 imaginary part) or occur in conjugate pairs

返回的w是其特征值。特征值不会特意进行排序。返回的array一般都是复数形式，除非虚部为0，会被cast为实数。当a是实数类型时，返回的就是实数。

v : (…, M, M) array

The normalized (unit “length”) eigenvectors, such that the column v[:,i] is the eigenvector corresponding to the eigenvalue w[i].

返回的v是归一化后的特征向量（length为1）。特征向量v[:,i]对应特征值w[i]。

Raises

LinAlgError

If the eigenvalue computation does not converge.

Ralated Function:

See Also

eigvals : eigenvalues of a non-symmetric array.
eigh : eigenvalues and eigenvectors of a real symmetric or complex Hermitian (conjugate symmetric) array.
eigvalsh : eigenvalues of a real symmetric or complex Hermitian (conjugate symmetric) array.
scipy.linalg.eig : Similar function in SciPy that also solves the generalized eigenvalue problem.
scipy.linalg.schur : Best choice for unitary and other non-Hermitian normal matrices.

Notes

… versionadded:: 1.8.0

Broadcasting rules apply, see the numpy.linalg documentation for details.

This is implemented using the _geev LAPACK routines which compute the eigenvalues and eigenvectors of general square arrays.

The number w is an eigenvalue of a if there exists a vector v such that a @ v = w * v. Thus, the arrays a, w, and v satisfy the equations a @ v[:,i] = w[i] * v[:,i] for :math:i \\in \\{0,...,M-1\\}.

The array v of eigenvectors may not be of maximum rank, that is, some of the columns may be linearly dependent, although round-off error may obscure that fact. If the eigenvalues are all different, then theoretically the eigenvectors are linearly independent and a can be diagonalized by a similarity transformation using v, i.e, inv(v) @ a @ v is diagonal.

For non-Hermitian normal matrices the SciPy function scipy.linalg.schur is preferred because the matrix v is guaranteed to be unitary, which is not the case when using eig. The Schur factorization produces an upper triangular matrix rather than a diagonal matrix, but for normal matrices only the diagonal of the upper triangular matrix is needed, the rest is roundoff error.

Finally, it is emphasized that v consists of the right (as in right-hand side) eigenvectors of a. A vector y satisfying y.T @ a = z * y.T for some number z is called a left eigenvector of a, and, in general, the left and right eigenvectors of a matrix are not necessarily the (perhaps conjugate) transposes of each other.
References
G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL,
Academic Press, Inc., 1980, Various pp.

需要说明的是，特征向量之间可能存在线性相关关系，即返回的v可能不是满秩的。但如果特征值都不同的话，理论上来说，所有特征向量都是线性无关的。

此时可以利用inv(v)@ a @ v来计算特征值的对角矩阵（对角线上的元素是特征值，其余元素为0),同时可以用v @ diag(w) @ inv(v)来恢复a。
同时需要说明的是，这里得到的特征向量都是右特征向量。

即 Ax=λx

Examples

>>> from numpy import linalg as LA

(Almost) trivial example with real e-values and e-vectors.

>>> w, v = LA.eig(np.diag((1, 2, 3)))
>>> w; v
array([1., 2., 3.])
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])

Real matrix possessing complex e-values and e-vectors; note that the
e-values are complex conjugates of each other.

>>> w, v = LA.eig(np.array([[1, -1], [1, 1]]))
>>> w; v
array([1.+1.j, 1.-1.j])
array([[0.70710678+0.j        , 0.70710678-0.j        ],
       [0.        -0.70710678j, 0.        +0.70710678j]])

Complex-valued matrix with real e-values (but complex-valued e-vectors);
note that ``a.conj().T == a``, i.e., `a` is Hermitian.

>>> a = np.array([[1, 1j], [-1j, 1]])
>>> w, v = LA.eig(a)
>>> w; v
array([2.+0.j, 0.+0.j])
array([[ 0.        +0.70710678j,  0.70710678+0.j        ], # may vary
       [ 0.70710678+0.j        , -0.        +0.70710678j]])

Be careful about round-off error!

>>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
>>> # Theor. e-values are 1 +/- 1e-9
>>> w, v = LA.eig(a)
>>> w; v
array([1., 1.])
array([[1., 0.],
       [0., 1.]])

总结

以上为个人经验，希望能给大家一个参考，也希望大家多多支持。