# numpy.linalg.eigh¶

`numpy.linalg.``eigh`(a, UPLO='L')[source]

Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix.

Returns two objects, a 1-D array containing the eigenvalues of a, and a 2-D square array or matrix (depending on the input type) of the corresponding eigenvectors (in columns).

Parameters: a : (…, M, M) array Hermitian or real symmetric matrices whose eigenvalues and eigenvectors are to be computed. UPLO : {‘L’, ‘U’}, optional Specifies whether the calculation is done with the lower triangular part of a (‘L’, default) or the upper triangular part (‘U’). Irrespective of this value only the real parts of the diagonal will be considered in the computation to preserve the notion of a Hermitian matrix. It therefore follows that the imaginary part of the diagonal will always be treated as zero. w : (…, M) ndarray The eigenvalues in ascending order, each repeated according to its multiplicity. v : {(…, M, M) ndarray, (…, M, M) matrix} The column `v[:, i]` is the normalized eigenvector corresponding to the eigenvalue `w[i]`. Will return a matrix object if a is a matrix object. LinAlgError If the eigenvalue computation does not converge.

`eigvalsh`
eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.
`eig`
eigenvalues and right eigenvectors for non-symmetric arrays.
`eigvals`
eigenvalues of non-symmetric arrays.

Notes

New in version 1.8.0.

Broadcasting rules apply, see the `numpy.linalg` documentation for details.

The eigenvalues/eigenvectors are computed using LAPACK routines `_syevd`, `_heevd`.

The eigenvalues of real symmetric or complex Hermitian matrices are always real. [1] The array v of (column) eigenvectors is unitary and a, w, and v satisfy the equations `dot(a, v[:, i]) = w[i] * v[:, i]`.

References

 [1] G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222.

Examples

```>>> from numpy import linalg as LA
>>> a = np.array([[1, -2j], [2j, 5]])
>>> a
array([[ 1.+0.j, -0.-2.j],
[ 0.+2.j,  5.+0.j]])
>>> w, v = LA.eigh(a)
>>> w; v
array([0.17157288, 5.82842712])
array([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
[ 0.        +0.38268343j,  0.        -0.92387953j]])
```
```>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j])
>>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/vec pair
array([0.+0.j, 0.+0.j])
```
```>>> A = np.matrix(a) # what happens if input is a matrix object
>>> A
matrix([[ 1.+0.j, -0.-2.j],
[ 0.+2.j,  5.+0.j]])
>>> w, v = LA.eigh(A)
>>> w; v
array([0.17157288, 5.82842712])
matrix([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
[ 0.        +0.38268343j,  0.        -0.92387953j]])
```
```>>> # demonstrate the treatment of the imaginary part of the diagonal
>>> a = np.array([[5+2j, 9-2j], [0+2j, 2-1j]])
>>> a
array([[5.+2.j, 9.-2.j],
[0.+2.j, 2.-1.j]])
>>> # with UPLO='L' this is numerically equivalent to using LA.eig() with:
>>> b = np.array([[5.+0.j, 0.-2.j], [0.+2.j, 2.-0.j]])
>>> b
array([[5.+0.j, 0.-2.j],
[0.+2.j, 2.+0.j]])
>>> wa, va = LA.eigh(a)
>>> wb, vb = LA.eig(b)
>>> wa; wb
array([1., 6.])
array([6.+0.j, 1.+0.j])
>>> va; vb
array([[-0.4472136 +0.j        , -0.89442719+0.j        ], # may vary
[ 0.        +0.89442719j,  0.        -0.4472136j ]])
array([[ 0.89442719+0.j       , -0.        +0.4472136j],
[-0.        +0.4472136j,  0.89442719+0.j       ]])
```

numpy.linalg.eig

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