eigvalsh

eigvalsh(x: array, /) → array

Returns the eigenvalues of a complex Hermitian or real symmetric matrix (or a stack of matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The eigenvalues of a complex Hermitian or real symmetric matrix $x\in {\mathbb{K}}^{n\times n}$ are defined as the roots (counted with multiplicity) of the polynomial $p$ of degree $n$ given by

$$p(\lambda )=\det (x-\lambda I_n)$$

where $\lambda \in \mathbb{R}$ and where $I_n$ is the n-dimensional identity matrix.

Note

Whether an array library explicitly checks whether an input array is Hermitian or a symmetric matrix (or a stack of matrices) is implementation-defined.

Note

The function eigvals will be added in a future version of the specification.

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

Returns:

out (array) – an array containing the computed eigenvalues. The returned array must have shape (..., M) and have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then must have a float64 data type).

Notes

Note

Eigenvalue sort order is left unspecified and is thus implementation-dependent.

Changed in version 2022.12: Added complex data type support.