scipy.linalg.eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False, overwrite_b=False, turbo=True, eigvals=None, type=1)[source]

Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

Find eigenvalues w and optionally eigenvectors v of matrix a, where b is positive definite:

              a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1
Parameters :

a : array, shape (M, M)

A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed.

b : array, shape (M, M)

A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.

lower : boolean

Whether the pertinent array data is taken from the lower or upper triangle of a. (Default: lower)

eigvals_only : boolean

Whether to calculate only eigenvalues and no eigenvectors. (Default: both are calculated)

turbo : boolean

Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None)

eigvals : tuple (lo, hi)

Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo < hi <= M-1. If omitted, all eigenvalues and eigenvectors are returned.

type: integer :

Specifies the problem type to be solved:

type = 1: a v[:,i] = w[i] b v[:,i] type = 2: a b v[:,i] = w[i] v[:,i] type = 3: b a v[:,i] = w[i] v[:,i]

overwrite_a : boolean

Whether to overwrite data in a (may improve performance)

overwrite_b : boolean

Whether to overwrite data in b (may improve performance)

Returns :

w : real array, shape (N,)

The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.

(if eigvals_only == False) :

v : complex array, shape (M, N)

The normalized selected eigenvector corresponding to the eigenvalue w[i] is the column v[:,i]. Normalization: type 1 and 3: v.conj() a v = w type 2: inv(v).conj() a inv(v) = w type = 1 or 2: v.conj() b v = I type = 3 : v.conj() inv(b) v = I

Raises LinAlgError if eigenvalue computation does not converge, :

an error occurred, or b matrix is not definite positive. Note that :

if input matrices are not symmetric or hermitian, no error is reported :

but results will be wrong. :

See also

eigenvalues and right eigenvectors for non-symmetric arrays

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