numpy求矩陣的特徵值與特徵向量(np.linalg.eig函數用法)

求矩陣的特徵值與特徵向量(np.linalg.eig)

語法

np.linalg.eig(a)

功能

Compute the eigenvalues and right eigenvectors of a square array.

求方陣(n x n)的特徵值與右特徵向量

Parameters

a : (…, M, M) array

Matrices for which the eigenvalues and right eigenvectors will be computed

a是一個矩陣Matrix的陣列。每個矩陣M都會被計算其特徵值與特徵向量。

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是其特徵值。特徵值不會特意進行排序。返回的array一般都是複數形式，除非虛部為0，會被cast為實數。當a是實數型別時，返回的就是實數。

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是歸一化後的特徵向量（length為1）。特徵向量v[:,i]對應特徵值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.

相關的函數有：

• eigvals：計算非對稱矩陣的特徵值
• eigh：實對稱矩陣或者複共軛對稱矩陣(Hermitian)的特徵值與特徵向量
• eigvalsh: 實對稱矩陣或者複共軛對稱矩陣(Hermitian)的特徵值與特徵向量
• scipy.linalg.eig
• scipy.linalg.schur

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.

需要說明的是，特徵向量之間可能存線上性相關關係，即返回的v可能不是滿秩的。但如果特徵值都不同的話，理論上來說，所有特徵向量都是線性無關的。

此時可以利用inv(v)@ a @ v來計算特徵值的對角矩陣（對角線上的元素是特徵值，其餘元素為0),同時可以用v @ diag(w) @ inv(v)來恢復a。
同時需要說明的是，這裡得到的特徵向量都是右特徵向量。

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.]])

總結

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