satie4

from fractions import Fraction
import sympy, numpy as np

def matrix_rang(A):
    return len([1 for i in A if any(i)])

def edin_matrix(n, m, A):
    for i in range(n):
        A[i] /= A[i][i]

    for i in range(n-1, -1, -1):
        for j in range(i-1, -1, -1):
            A[j] += -A[j][i]*A[i]

    return A

def svobodnie_znach(n, m, A):
    res = set()
    for i in range(n):
        for j in range(m):
            if A[i][j] != 0:
                res.add(j)
                break
    return set(range(m)).difference(res)

def stupen_matrix(n, m, A):
    for x in range(n):
        y = x
        if x < n and y < m and not A[x][y] or not any(A[x]):
            A = np.append(A, [A[x]], axis=0)
            A = np.delete(A, x, axis=0)
        for i in range(1+x, n):
            while y < m-1 and not A[x][y]:
                if A[x][y]:
                    break
                y += 1
            if y == m or not A[x][y]:
                continue

            K = -A[i][y]/A[x][y]
            for j in range(m):
                A[i][j] = A[x][j]*K + A[i][j]

    for x in range(n):
        if not any(A[x]):
            A = np.append(A, [A[x]], axis=0)
            A = np.delete(A, x, axis=0)
    return A

def fsr(n, m, A, basisnie, svobodnie):
    k = len(basisnie)
    ans = np.array([m*[Fraction(0.)] for x in range(len(svobodnie))])
    for ind, x in enumerate(basisnie):
        for i, elem in enumerate(A[ind][k:]):
            if elem != 0:
                ans[i][x] = elem
    for i, x in enumerate(svobodnie):
        ans[i][x] = 1
    return ans

def reshit_slau(n, m, A):
    A = stupen_matrix(n, m, A)
    A = A[:matrix_rang(A)]
    n = len(A)
    svobodnie = list(svobodnie_znach(n, m, A))
    basisnie = list(set(range(m)).difference(set(svobodnie)))
    a = [[] for x in range(len(A))]
    for i in sorted(svobodnie, reverse=True):
        a = np.append(a, -1*A[:, i:i+1], axis=1)
        A = np.delete(A, i, axis=1)
    A = np.append(A, a[:, ::-1], axis=1)
    return fsr(n, m, edin_matrix(n, m, A), basisnie, svobodnie)

def main():
    var = sympy.Symbol('x')
    n = int(input())
    A = sympy.Matrix([[Fraction(x) for x in input().split()] for i in range(n)])
    B = sympy.matrices.eye(n, n)*var
    print('\nСоставляем характеристическое уравнение:')
    sympy.pprint(A-B)
    print()
    eq = sympy.Eq(sympy.det(A-B), 0)
    print('det(A − x*E)=')
    sympy.pprint(eq)
    print()
    roots = sympy.roots(eq, var)
    print('Собственные значения и их кратности:')
    print(roots, '\n')
    print('Матрица линейного оператора A в базисе, состоящем из собственных векторов:')
    sympy.pprint(sympy.matrices.diag(*[x for x, i in sorted(roots.items()) for y in range(i)]))
    print()

    result = sympy.Matrix([])
    for step, root in enumerate(sorted(roots.keys()), start=1):
        # Находим собственные вектора
        a = reshit_slau(n, n, np.array(A - sympy.matrices.eye(n, n)*root))
        # Ортонормируем собственные вектора
        a = sympy.matrices.GramSchmidt(reversed([sympy.Matrix(i) for i in a]))
        a = [sympy.Matrix(i)/sympy.Matrix(i).norm() for i in a]
        for i in a:
            result = result.col_insert(step, sympy.Matrix(i))

    print('Ортнормированный базис')
    sympy.pprint(result)

    # print('\nПроверка')
    # print('S^-1 * A * S = A(s)')
    # sympy.pprint([result.T, '*', A, '*', result, '=', result.T*A*result])

if __name__ == '__main__':
    main()