numpy求矩陣的特徵值與特徵向量(np.linalg.eig函數用法)

2023-02-06 06:01:29

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.

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

v : (…, M, M) array

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

Raises

LinAlgError

If the eigenvalue computation does not converge.

Ralated Function:

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.

• 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

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.

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]])

>>> 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.]])



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