Source code for cla_utils.exercises8

import numpy as np

def Q1AQ1s(A):
    """
    For a matrix A, find the unitary matrix Q1 such that the first
    column of Q1*A has zeros below the diagonal. Then return A1 = Q1*A*Q1^*.

    :param A: an mxm numpy array

    :return A1: an mxm numpy array
    """

    raise NotImplementedError


def hessenberg(A):
    """
    For a matrix A, transform to Hessenberg form H by Householder
    similarity transformations, in place.

    :param A: an mxm numpy array
    """

    raise NotImplementedError


def hessenbergQ(A):
    """
    For a matrix A, transform to Hessenberg form H by Householder
    similarity transformations, in place, and return the matrix Q
    for which QHQ^* = A.

    :param A: an mxm numpy array
    
    :return Q: an mxm numpy array
    """

    raise NotImplementedError

def hessenberg_ev(H):
    """
    Given a Hessenberg matrix, return the eigenvectors.

    :param H: an mxm numpy array

    :return V: an mxm numpy array whose columns are the eigenvectors of H

    Do not change this function.
    """
    m, n = H.shape
    assert(m==n)
    assert(cla_utils.norm(H[np.tril_indices(m, -2)]) < 1.0e-6)
    _, V = np.linalg.eig(H)
    return V


def ev(A):
    """
    Given a matrix A, return the eigenvectors of A. This should
    be done by using your functions to reduce to upper Hessenberg
    form, before calling hessenberg_ev (which you should not edit!).

    :param A: an mxm numpy array

    :return V: an mxm numpy array whose columns are the eigenvectors of A
    """

    raise NotImplementedError