
"""simple docstring"""




from typing import Any

import numpy as np



def     __magic_name__		(					UpperCamelCase     :		np.ndarray     )     ->						bool:
    return np.array_equal(UpperCamelCase							,		matrix.conjugate().T     )



def     __magic_name__		(					UpperCamelCase     :		np.ndarray							,		UpperCamelCase     :		np.ndarray     )     ->						Any:
    a__		=			v.conjugate().T
    a__		=			v_star.dot(UpperCamelCase     )
    assert isinstance(UpperCamelCase							,		np.ndarray     )
    return (v_star_dot.dot(UpperCamelCase     )) / (v_star.dot(UpperCamelCase     ))



def     __magic_name__		(					)     ->						None:
    a__		=			np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]     )
    a__		=			np.array([[1], [2], [3]]     )
    assert is_hermitian(UpperCamelCase     ), f'{a} is not hermitian.'
    print(rayleigh_quotient(UpperCamelCase							,		UpperCamelCase     )     )

    a__		=			np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]     )
    assert is_hermitian(UpperCamelCase     ), f'{a} is not hermitian.'
    assert rayleigh_quotient(UpperCamelCase							,		UpperCamelCase     ) == float(3     )


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    tests()




"""simple docstring"""




from __future__ import annotations



def     __magic_name__		(					UpperCamelCase     :		str							,		UpperCamelCase     :		list[str] | None = None     )     ->						list[list[str]]:
    a__		=			word_bank or []
    # create a table
    a__		=			len(UpperCamelCase     ) + 1

    a__		=			[]
    for _ in range(UpperCamelCase     ):
        table.append([]     )
    # seed value
    a__		=			[[]]  # because empty string has empty combination

    # iterate through the indices
    for i in range(UpperCamelCase     ):
        # condition
        if table[i] != []:
            for word in word_bank:
                # slice condition
                if target[i : i + len(UpperCamelCase     )] == word:
                    a__		=			[
                        [word, *way] for way in table[i]
                    ]
                    # adds the word to every combination the current position holds
                    # now,push that combination to the table[i+len(word)]
                    table[i + len(UpperCamelCase     )] += new_combinations

    # combinations are in reverse order so reverse for better output
    for combination in table[len(UpperCamelCase     )]:
        combination.reverse()

    return table[len(UpperCamelCase     )]


if __name__ == "__main__":
    print(all_construct('jwajalapa', ['jwa', 'j', 'w', 'a', 'la', 'lapa']))
    print(all_construct('rajamati', ['s', 'raj', 'amat', 'raja', 'ma', 'i', 't']))
    print(
        all_construct(
            'hexagonosaurus',
            ['h', 'ex', 'hex', 'ag', 'ago', 'ru', 'auru', 'rus', 'go', 'no', 'o', 's'],
        )
    )


