canon_basis_problem4

from fractions import Fraction as f

import sympy as s
from sympy import pprint
from new_test2 import GramGauss


var = s.Symbol('λ')
n = int(input())
a = s.Matrix([[f(x) for x in input().split()] for i in range(n)])
b = s.matrices.eye(n, n)*var
pprint(b)
print()
pprint(a-b)
print()
equation = s.Eq(s.det(a-b), 0)
pprint(equation)
ans = s.roots(equation, var)
print(ans, '\n')
# Матрица вида 9.20
pprint(s.matrices.diag(*[x for x, i in sorted(ans.items()) for _ in range(i)]))
print()

result = s.Matrix([])
for step, root in enumerate(sorted(ans.keys())):
    res = a - s.matrices.eye(n, n)*root
    # ФСР
    res = GramGauss(n, n, res).silent_solve()
    # Ортогонализация
    res = s.matrices.GramSchmidt([s.Matrix(i) for i in res])
    res = [s.Matrix(i)/s.Matrix(i).norm() for i in res]
    for i in res:
        result = result.col_insert(step+1, s.Matrix(i))

print('Ортонормированный базис')
pprint(result)

print('Проверка')
pprint(result.T*a*result)