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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
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.
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
New in version 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 (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(.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(.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 = .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]])
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