nlcpy.linalg.eig

nlcpy.linalg.eig(a)[source]

Computes the eigenvalues and right eigenvectors of a square array.

Parameters

a(…, M, M) array_like: Matrices for which the eigenvalues and right eigenvectors will be computed.

Returns

w(…, M) ndarray: 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. Please note that there are cases where rounding errors affect the dtype of the array. When a is real the resulting eigenvalues will be real (0 imaginary part) or occur in conjugate pairs.
v(…, M, M) ndarray: The normalized (unit “length”) eigenvectors, such that the column v[:,i] is the eigenvector corresponding to the eigenvalue w[i].

See also

eigvals: Computes the eigenvalues of a general matrix.
eigh: Computes the eigenvalues and eigenvectors of a complex Hermitian or a real symmetric matrix.
eigvalsh: Computes the eigenvalues of a complex Hermitian or real symmetric matrix.

Note

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, 1, …, 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. Likewise, the (complex-valued) matrix of eigenvectors v is unitary if the matrix a is normal, i.e., if dot(a, a.H) = dot(a.H, a), where a.H denotes the conjugate transpose of a.

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.

Examples

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

>>> import nlcpy as vp
>>> w, v = vp.linalg.eig(vp.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 = vp.linalg.eig(vp.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 = vp.array([[1, 1j], [-1j, 1]])
>>> w, v = vp.linalg.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 = vp.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
>>> # Theor. e-values are 1 +/- 1e-9
>>> w, v = vp.linalg.eig(a)
>>> w; v   
array([1., 1.])
array([[1., 0.],
       [0., 1.]])