def det(Matrix): m = len(Matrix) if m == 1: return Matrix[0][0] sign = 1 res = 0 for j in range(m): M = [[0] * (m - 1) for i in range(m - 1)] for i in range(1, m): k = 0 for l in range(m): if l != j: M[i-1][k] = Matrix[i][l] k += 1 res += sign * Matrix[0][j] * det(M) sign = -sign return res逐行解释

这是一个计算矩阵行列式的函数，输入参数为一个矩阵Matrix。首先获取矩阵的行数m，如果矩阵只有一个元素，则直接返回该元素。接下来定义一个符号变量sign和结果变量res，初始值都为0。然后遍历矩阵的每一列，对于每一列，都构造一个(m-1)×(m-1)的子矩阵M，方法是将该列去掉，然后将剩余的元素填入子矩阵中。接着递归调用det函数，计算子矩阵的行列式值，将结果乘以该列的元素值和符号sign，然后累加到结果变量res中。最后返回结果变量res即可。

相关问题

def mat_simp(M): n, p = M.shape for i in range(n): for j in range(p): M[i, j] = M[i, j].simplify().expand().simplify() return M def mat(u): n = len(u) M = sp.zeros(n, n) for j in range(n): cj = sp.Matrix([int(i == j) for i in range(n)]) Mcj = cj-2*cj.dot(u)/u.norm()**2*u for i in range(n): M[i, j] = Mcj[i] return M def DecompositionQR(A): n, p = A.shape if n != p or A.det() == 0: raise Exception("Les données ne sont pas correctes") AA = A.copy() O = sp.eye(n) for i in range(n): if not AA[i+1:, i].is_zero_matrix: B = AA.copy() for j in range(i): B[j, i] = 0 Ci = sp.Matrix([int(k == i) for k in range(n)]) S = mat_simp(mat(Ci-B[:, i]/B[:, i].norm())) O = S@O AA = S@AA return O.transpose(), AA修改这段代码，使她可以做QR分解

以下是修改后的代码，可以进行 QR 分解：

def mat_simp(M):
    n, p = M.shape
    for i in range(n):
        for j in range(p):
            M[i, j] = M[i, j].simplify().expand().simplify()
    return M

def mat(u):
    n = len(u)
    M = sp.zeros(n, n)
    for j in range(n):
        cj = sp.Matrix([int(i == j) for i in range(n)])
        Mcj = cj-2*cj.dot(u)/u.norm()**2*u
        for i in range(n):
            M[i, j] = Mcj[i]
    return M

def qr_decomposition(A):
    n, p = A.shape
    if n != p or A.det() == 0:
        raise Exception("Les données ne sont pas correctes")
    AA = A.copy()
    Q = sp.eye(n)
    for i in range(n):
        if not AA[i+1:, i].is_zero_matrix:
            B = AA.copy()
            for j in range(i):
                B[j, i] = 0
            Ci = sp.Matrix([int(k == i) for k in range(n)])
            S = mat_simp(mat(Ci-B[:, i]/B[:, i].norm()))
            Q = Q@S
            AA = S@AA
    R = AA
    return Q.transpose(), R

将函数名 `DecompositionQR` 改成了 `qr_decomposition`，并且修改了函数主体部分，使其能够进行 QR 分解。

def mat_simp(A): n, p = A.shape for i in range(n): for j in range(p): A[i, j] = A[i, j].simplify().expand().simplify() return A def mat(u): n = len(u) M = sp.zeros(n, n) for j in range(n): cj = sp.Matrix([int(i == j) for i in range(n)]) Mcj = cj-2*cj.dot(u)/u.norm()**2*u for i in range(n): M[i, j] = Mcj[i] return M def DecompositionQR(A): n, p = A.shape if n != p or A.det() == 0: raise Exception("Les données ne sont pas correctes") AA = A.copy() O = sp.eye(n) for i in range(n): if not AA[i+1:, i].is_zero_matrix: B = AA.copy() for j in range(i): B[j, i] = 0 Ci = sp.Matrix([int(k == i) for k in range(n)]) S = mat_simp(mat(Ci-B[:, i]/B[:, i].norm())) O = S@O AA = S@AA return O.transpose(), AA给这段代码写上法语注释

def mat_simp(A):
# Simplifie la matrice A
n, p = A.shape
for i in range(n):
for j in range(p):
A[i, j] = A[i, j].simplify().expand().simplify()
return A

def mat(u):
# Calcule la matrice M correspondant à un vecteur u donné
n = len(u)
M = sp.zeros(n, n)
for j in range(n):
cj = sp.Matrix([int(i == j) for i in range(n)])
Mcj = cj-2*cj.dot(u)/u.norm()**2*u
for i in range(n):
M[i, j] = Mcj[i]
return M

def DecompositionQR(A):
# Décompose la matrice A en Q et R
n, p = A.shape
if n != p or A.det() == 0:
raise Exception("Les données ne sont pas correctes")
AA = A.copy()
O = sp.eye(n)
for i in range(n):
if not AA[i+1:, i].is_zero_matrix:
B = AA.copy()
for j in range(i):
B[j, i] = 0
Ci = sp.Matrix([int(k == i) for k in range(n)])
S = mat_simp(mat(Ci-B[:, i]/B[:, i].norm()))
O = S@O
AA = S@AA
return O.transpose(), AA

# Les commentaires sont ajoutés en français pour expliquer les fonctions et les variables.