Multiplication de matrices en python?
J'essaie de multiplier deux matrices ensemble en utilisant du python pur. Entrée (X1 est un 3x3 et Xt est un 3x2):
X1 = [[1.0016, 0.0, -16.0514],
[0.0, 10000.0, -40000.0],
[-16.0514, -40000.0, 160513.6437]]
Xt = [(1.0, 1.0),
(0.0, 0.25),
(0.0, 0.0625)]
où Xt est la transposée Zip d'une autre matrice. Maintenant, voici le code:
def matrixmult (A, B):
C = [[0 for row in range(len(A))] for col in range(len(B[0]))]
for i in range(len(A)):
for j in range(len(B[0])):
for k in range(len(B)):
C[i][j] += A[i][k]*B[k][j]
return C
L'erreur que me donne python est la suivante: IndexError: index de liste hors limites. Maintenant, je ne sais pas si Xt est reconnu en tant que matrice et reste un objet de liste, mais techniquement, cela devrait fonctionner.
Si vous ne voulez vraiment pas utiliser numpy, vous pouvez faire quelque chose comme ceci:
def matmult(a,b):
Zip_b = Zip(*b)
# uncomment next line if python 3 :
# Zip_b = list(Zip_b)
return [[sum(ele_a*ele_b for ele_a, ele_b in Zip(row_a, col_b))
for col_b in Zip_b] for row_a in a]
x = [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
y = [[1,2],[1,2],[3,4]]
import numpy as np # I want to check my solution with numpy
mx = np.matrix(x)
my = np.matrix(y)
Résultat:
>>> matmult(x,y)
[[12, 18], [27, 42], [42, 66], [57, 90]]
>>> mx * my
matrix([[12, 18],
[27, 42],
[42, 66],
[57, 90]])
Ceci est une initialisation incorrecte. Vous avez échangé la ligne avec col!
C = [[0 for row in range(len(A))] for col in range(len(B[0]))]
Une initialisation correcte serait
C = [[0 for col in range(len(B[0]))] for row in range(len(A))]
Aussi, je suggérerais d'utiliser de meilleures conventions de nommage. Vous aidera beaucoup dans le débogage. Par exemple:
def matrixmult (A, B):
rows_A = len(A)
cols_A = len(A[0])
rows_B = len(B)
cols_B = len(B[0])
if cols_A != rows_B:
print "Cannot multiply the two matrices. Incorrect dimensions."
return
# Create the result matrix
# Dimensions would be rows_A x cols_B
C = [[0 for row in range(cols_B)] for col in range(rows_A)]
print C
for i in range(rows_A):
for j in range(cols_B):
for k in range(cols_A):
C[i][j] += A[i][k] * B[k][j]
return C
Vous pouvez faire beaucoup plus, mais vous avez l'idée ...
Voici un code court et simple pour les routines matrice/vecteur en pur Python que j'ai écrit il y a de nombreuses années:
'''Basic Table, Matrix and Vector functions for Python 2.2
Author: Raymond Hettinger
'''
Version = 'File MATFUNC.PY, Ver 183, Date 12-Dec-2002,14:33:42'
import operator, math, random
NPRE, NPOST = 0, 0 # Disables pre and post condition checks
def iszero(z): return abs(z) < .000001
def getreal(z):
try:
return z.real
except AttributeError:
return z
def getimag(z):
try:
return z.imag
except AttributeError:
return 0
def getconj(z):
try:
return z.conjugate()
except AttributeError:
return z
separator = [ '', '\t', '\n', '\n----------\n', '\n===========\n' ]
class Table(list):
dim = 1
concat = list.__add__ # A substitute for the overridden __add__ method
def __getslice__( self, i, j ):
return self.__class__( list.__getslice__(self,i,j) )
def __init__( self, elems ):
list.__init__( self, elems )
if len(elems) and hasattr(elems[0], 'dim'): self.dim = elems[0].dim + 1
def __str__( self ):
return separator[self.dim].join( map(str, self) )
def map( self, op, rhs=None ):
'''Apply a unary operator to every element in the matrix or a binary operator to corresponding
elements in two arrays. If the dimensions are different, broadcast the smaller dimension over
the larger (i.e. match a scalar to every element in a vector or a vector to a matrix).'''
if rhs is None: # Unary case
return self.dim==1 and self.__class__( map(op, self) ) or self.__class__( [elem.map(op) for elem in self] )
Elif not hasattr(rhs,'dim'): # List / Scalar op
return self.__class__( [op(e,rhs) for e in self] )
Elif self.dim == rhs.dim: # Same level Vec / Vec or Matrix / Matrix
assert NPRE or len(self) == len(rhs), 'Table operation requires len sizes to agree'
return self.__class__( map(op, self, rhs) )
Elif self.dim < rhs.dim: # Vec / Matrix
return self.__class__( [op(self,e) for e in rhs] )
return self.__class__( [op(e,rhs) for e in self] ) # Matrix / Vec
def __mul__( self, rhs ): return self.map( operator.mul, rhs )
def __div__( self, rhs ): return self.map( operator.div, rhs )
def __sub__( self, rhs ): return self.map( operator.sub, rhs )
def __add__( self, rhs ): return self.map( operator.add, rhs )
def __rmul__( self, lhs ): return self*lhs
def __rdiv__( self, lhs ): return self*(1.0/lhs)
def __rsub__( self, lhs ): return -(self-lhs)
def __radd__( self, lhs ): return self+lhs
def __abs__( self ): return self.map( abs )
def __neg__( self ): return self.map( operator.neg )
def conjugate( self ): return self.map( getconj )
def real( self ): return self.map( getreal )
def imag( self ): return self.map( getimag )
def flatten( self ):
if self.dim == 1: return self
return reduce( lambda cum, e: e.flatten().concat(cum), self, [] )
def prod( self ): return reduce(operator.mul, self.flatten(), 1.0)
def sum( self ): return reduce(operator.add, self.flatten(), 0.0)
def exists( self, predicate ):
for elem in self.flatten():
if predicate(elem):
return 1
return 0
def forall( self, predicate ):
for elem in self.flatten():
if not predicate(elem):
return 0
return 1
def __eq__( self, rhs ): return (self - rhs).forall( iszero )
class Vec(Table):
def dot( self, otherVec ): return reduce(operator.add, map(operator.mul, self, otherVec), 0.0)
def norm( self ): return math.sqrt(abs( self.dot(self.conjugate()) ))
def normalize( self ): return self / self.norm()
def outer( self, otherVec ): return Mat([otherVec*x for x in self])
def cross( self, otherVec ):
'Compute a Vector or Cross Product with another vector'
assert len(self) == len(otherVec) == 3, 'Cross product only defined for 3-D vectors'
u, v = self, otherVec
return Vec([ u[1]*v[2]-u[2]*v[1], u[2]*v[0]-u[0]*v[2], u[0]*v[1]-u[1]*v[0] ])
def house( self, index ):
'Compute a Householder vector which zeroes all but the index element after a reflection'
v = Vec( Table([0]*index).concat(self[index:]) ).normalize()
t = v[index]
sigma = 1.0 - t**2
if sigma != 0.0:
t = v[index] = t<=0 and t-1.0 or -sigma / (t + 1.0)
v /= t
return v, 2.0 * t**2 / (sigma + t**2)
def polyval( self, x ):
'Vec([6,3,4]).polyval(5) evaluates to 6*x**2 + 3*x + 4 at x=5'
return reduce( lambda cum,c: cum*x+c, self, 0.0 )
def ratval( self, x ):
'Vec([10,20,30,40,50]).ratfit(5) evaluates to (10*x**2 + 20*x + 30) / (40*x**2 + 50*x + 1) at x=5.'
degree = len(self) / 2
num, den = self[:degree+1], self[degree+1:] + [1]
return num.polyval(x) / den.polyval(x)
class Matrix(Table):
__slots__ = ['size', 'rows', 'cols']
def __init__( self, elems ):
'Form a matrix from a list of lists or a list of Vecs'
Table.__init__( self, hasattr(elems[0], 'dot') and elems or map(Vec,map(Tuple,elems)) )
self.size = self.rows, self.cols = len(elems), len(elems[0])
def tr( self ):
'Tranpose elements so that Transposed[i][j] = Original[j][i]'
return Mat(Zip(*self))
def star( self ):
'Return the Hermetian adjoint so that Star[i][j] = Original[j][i].conjugate()'
return self.tr().conjugate()
def diag( self ):
'Return a vector composed of elements on the matrix diagonal'
return Vec( [self[i][i] for i in range(min(self.size))] )
def trace( self ): return self.diag().sum()
def mmul( self, other ):
'Matrix multiply by another matrix or a column vector '
if other.dim==2: return Mat( map(self.mmul, other.tr()) ).tr()
assert NPRE or self.cols == len(other)
return Vec( map(other.dot, self) )
def augment( self, otherMat ):
'Make a new matrix with the two original matrices laid side by side'
assert self.rows == otherMat.rows, 'Size mismatch: %s * %s' % (`self.size`, `otherMat.size`)
return Mat( map(Table.concat, self, otherMat) )
def qr( self, ROnly=0 ):
'QR decomposition using Householder reflections: Q*R==self, Q.tr()*Q==I(n), R upper triangular'
R = self
m, n = R.size
for i in range(min(m,n)):
v, beta = R.tr()[i].house(i)
R -= v.outer( R.tr().mmul(v)*beta )
for i in range(1,min(n,m)): R[i][:i] = [0] * i
R = Mat(R[:n])
if ROnly: return R
Q = R.tr().solve(self.tr()).tr() # Rt Qt = At nn nm = nm
self.qr = lambda r=0, c=`self`: not r and c==`self` and (Q,R) or Matrix.qr(self,r) #Cache result
assert NPOST or m>=n and Q.size==(m,n) and isinstance(R,UpperTri) or m<n and Q.size==(m,m) and R.size==(m,n)
assert NPOST or Q.mmul(R)==self and Q.tr().mmul(Q)==eye(min(m,n))
return Q, R
def _solve( self, b ):
'''General matrices (incuding) are solved using the QR composition.
For inconsistent cases, returns the least squares solution'''
Q, R = self.qr()
return R.solve( Q.tr().mmul(b) )
def solve( self, b ):
'Divide matrix into a column vector or matrix and iterate to improve the solution'
if b.dim==2: return Mat( map(self.solve, b.tr()) ).tr()
assert NPRE or self.rows == len(b), 'Matrix row count %d must match vector length %d' % (self.rows, len(b))
x = self._solve( b )
diff = b - self.mmul(x)
maxdiff = diff.dot(diff)
for i in range(10):
xnew = x + self._solve( diff )
diffnew = b - self.mmul(xnew)
maxdiffnew = diffnew.dot(diffnew)
if maxdiffnew >= maxdiff: break
x, diff, maxdiff = xnew, diffnew, maxdiffnew
#print >> sys.stderr, i+1, maxdiff
assert NPOST or self.rows!=self.cols or self.mmul(x) == b
return x
def rank( self ): return Vec([ not row.forall(iszero) for row in self.qr(ROnly=1) ]).sum()
class Square(Matrix):
def lu( self ):
'Factor a square matrix into lower and upper triangular form such that L.mmul(U)==A'
n = self.rows
L, U = eye(n), Mat(self[:])
for i in range(n):
for j in range(i+1,U.rows):
assert U[i][i] != 0.0, 'LU requires non-zero elements on the diagonal'
L[j][i] = m = 1.0 * U[j][i] / U[i][i]
U[j] -= U[i] * m
assert NPOST or isinstance(L,LowerTri) and isinstance(U,UpperTri) and L*U==self
return L, U
def __pow__( self, exp ):
'Raise a square matrix to an integer power (i.e. A**3 is the same as A.mmul(A.mmul(A))'
assert NPRE or exp==int(exp) and exp>0, 'Matrix powers only defined for positive integers not %s' % exp
if exp == 1: return self
if exp&1: return self.mmul(self ** (exp-1))
sqrme = self ** (exp/2)
return sqrme.mmul(sqrme)
def det( self ): return self.qr( ROnly=1 ).det()
def inverse( self ): return self.solve( eye(self.rows) )
def hessenberg( self ):
'''Householder reduction to Hessenberg Form (zeroes below the diagonal)
while keeping the same eigenvalues as self.'''
for i in range(self.cols-2):
v, beta = self.tr()[i].house(i+1)
self -= v.outer( self.tr().mmul(v)*beta )
self -= self.mmul(v).outer(v*beta)
return self
def eigs( self ):
'Estimate principal eigenvalues using the QR with shifts method'
origTrace, origDet = self.trace(), self.det()
self = self.hessenberg()
eigvals = Vec([])
for i in range(self.rows-1,0,-1):
while not self[i][:i].forall(iszero):
shift = eye(i+1) * self[i][i]
q, r = (self - shift).qr()
self = r.mmul(q) + shift
eigvals.append( self[i][i] )
self = Mat( [self[r][:i] for r in range(i)] )
eigvals.append( self[0][0] )
assert NPOST or iszero( (abs(origDet) - abs(eigvals.prod())) / 1000.0 )
assert NPOST or iszero( origTrace - eigvals.sum() )
return Vec(eigvals)
class Triangular(Square):
def eigs( self ): return self.diag()
def det( self ): return self.diag().prod()
class UpperTri(Triangular):
def _solve( self, b ):
'Solve an upper triangular matrix using backward substitution'
x = Vec([])
for i in range(self.rows-1, -1, -1):
assert NPRE or self[i][i], 'Backsub requires non-zero elements on the diagonal'
x.insert(0, (b[i] - x.dot(self[i][i+1:])) / self[i][i] )
return x
class LowerTri(Triangular):
def _solve( self, b ):
'Solve a lower triangular matrix using forward substitution'
x = Vec([])
for i in range(self.rows):
assert NPRE or self[i][i], 'Forward sub requires non-zero elements on the diagonal'
x.append( (b[i] - x.dot(self[i][:i])) / self[i][i] )
return x
def Mat( elems ):
'Factory function to create a new matrix.'
m, n = len(elems), len(elems[0])
if m != n: return Matrix(elems)
if n <= 1: return Square(elems)
for i in range(1, len(elems)):
if not iszero( max(map(abs, elems[i][:i])) ):
break
else: return UpperTri(elems)
for i in range(0, len(elems)-1):
if not iszero( max(map(abs, elems[i][i+1:])) ):
return Square(elems)
return LowerTri(elems)
def funToVec( tgtfun, low=-1, high=1, steps=40, EqualSpacing=0 ):
'''Compute x,y points from evaluating a target function over an interval (low to high)
at evenly spaces points or with Chebyshev abscissa spacing (default) '''
if EqualSpacing:
h = (0.0+high-low)/steps
xvec = [low+h/2.0+h*i for i in range(steps)]
else:
scale, base = (0.0+high-low)/2.0, (0.0+high+low)/2.0
xvec = [base+scale*math.cos(((2*steps-1-2*i)*math.pi)/(2*steps)) for i in range(steps)]
yvec = map(tgtfun, xvec)
return Mat( [xvec, yvec] )
def funfit( (xvec, yvec), basisfuns ):
'Solves design matrix for approximating to basis functions'
return Mat([ map(form,xvec) for form in basisfuns ]).tr().solve(Vec(yvec))
def polyfit( (xvec, yvec), degree=2 ):
'Solves Vandermonde design matrix for approximating polynomial coefficients'
return Mat([ [x**n for n in range(degree,-1,-1)] for x in xvec ]).solve(Vec(yvec))
def ratfit( (xvec, yvec), degree=2 ):
'Solves design matrix for approximating rational polynomial coefficients (a*x**2 + b*x + c)/(d*x**2 + e*x + 1)'
return Mat([[x**n for n in range(degree,-1,-1)]+[-y*x**n for n in range(degree,0,-1)] for x,y in Zip(xvec,yvec)]).solve(Vec(yvec))
def genmat(m, n, func):
if not n: n=m
return Mat([ [func(i,j) for i in range(n)] for j in range(m) ])
def zeroes(m=1, n=None):
'Zero matrix with side length m-by-m or m-by-n.'
return genmat(m,n, lambda i,j: 0)
def eye(m=1, n=None):
'Identity matrix with side length m-by-m or m-by-n'
return genmat(m,n, lambda i,j: i==j)
def hilb(m=1, n=None):
'Hilbert matrix with side length m-by-m or m-by-n. Elem[i][j]=1/(i+j+1)'
return genmat(m,n, lambda i,j: 1.0/(i+j+1.0))
def Rand(m=1, n=None):
'Random matrix with side length m-by-m or m-by-n'
return genmat(m,n, lambda i,j: random.random())
if __== '__main__':
import cmath
a = Table([1+2j,2,3,4])
b = Table([5,6,7,8])
C = Table([a,b])
print 'a+b', a+b
print '2+a', 2+a
print 'a/5.0', a/5.0
print '2*a+3*b', 2*a+3*b
print 'a+C', a+C
print '3+C', 3+C
print 'C+b', C+b
print 'C.sum()', C.sum()
print 'C.map(math.cos)', C.map(cmath.cos)
print 'C.conjugate()', C.conjugate()
print 'C.real()', C.real()
print zeroes(3)
print eye(4)
print hilb(3,5)
C = Mat( [[1,2,3], [4,5,1,], [7,8,9]] )
print C.mmul( C.tr()), '\n'
print C ** 5, '\n'
print C + C.tr(), '\n'
A = C.tr().augment( Mat([[10,11,13]]).tr() ).tr()
q, r = A.qr()
assert q.mmul(r) == A
assert q.tr().mmul(q)==eye(3)
print 'q:\n', q, '\nr:\n', r, '\nQ.tr()&Q:\n', q.tr().mmul(q), '\nQ*R\n', q.mmul(r), '\n'
b = Vec([50, 100, 220, 321])
x = A.solve(b)
print 'x: ', x
print 'b: ', b
print 'Ax: ', A.mmul(x)
inv = C.inverse()
print '\ninverse C:\n', inv, '\nC * inv(C):\n', C.mmul(inv)
assert C.mmul(inv) == eye(3)
points = (xvec,yvec) = funToVec(lambda x: math.sin(x)+2*math.cos(.7*x+.1), low=0, high=3, EqualSpacing=1)
basis = [lambda x: math.sin(x), lambda x: math.exp(x), lambda x: x**2]
print 'Func coeffs:', funfit( points, basis )
print 'Poly coeffs:', polyfit( points, degree=5 )
points = (xvec,yvec) = funToVec(lambda x: math.sin(x)+2*math.cos(.7*x+.1), low=0, high=3)
print 'Rational coeffs:', ratfit( points )
print polyfit(([1,2,3,4], [1,4,9,16]), 2)
mtable = Vec([1,2,3]).outer(Vec([1,2]))
print mtable, mtable.size
A = Mat([ [2,0,3], [1,5,1], [18,0,6] ])
print 'A:'
print A
print 'eigs:'
print A.eigs()
print 'Should be:', Vec([11.6158, 5.0000, -3.6158])
print 'det(A)'
print A.det()
c = Mat( [[1,2,30],[4,5,10],[10,80,9]] ) # Failed example from Konrad Hinsen
print 'C:\n', c
print c.eigs()
print 'Should be:', Vec([-8.9554, 43.2497, -19.2943])
A = Mat([ [1,2,3,4], [4,5,6,7], [2,1,5,0], [4,2,1,0] ] ) # Kincaid and Cheney p.326
print 'A:\n', A
print A.eigs()
print 'Should be:', Vec([3.5736, 0.1765, 11.1055, -3.8556])
A = Rand(3)
q,r = A.qr()
s,t = A.qr()
print q is s # Test caching
print r is t
A[1][1] = 1.1 # Invalidate the cache
u,v = A.qr()
print q is u # Verify old result not used
print r is v
print u.mmul(v) == A # Verify new result
print 'Test qr on 3x5 matrix'
a = Rand(3,5)
q,r = a.qr()
print q.mmul(r) == a
print q.tr().mmul(q) == eye(3)
En retard pour la fête, mais quand je devais faire de l'arithmétique matricielle, j'ai défini une nouvelle classe pour aider. Dans une telle classe, vous pouvez définir des méthodes telles que __add__ ou, dans votre cas d'utilisation, __matmul__, vous permettant de créer a * b ou a *= b plutôt que matrixMult(a,b).
J'ai inclus un code qui implémente ceci ci-dessous (j'ai exclu la méthode prohibitive __init__, qui crée essentiellement une liste bidimensionnelle self.mat et un tuple self.order en fonction de ce qui lui est transmis)
class matrix:
def __matmul__(self, multiplier):
if self.order[1] != multiplier.order[0]:
raise ValueError("The multiplier was non-conformable under multiplication.")
return [[sum(a*b for a,b in Zip(srow,mcol)) for mcol in Zip(*multiplier.mat)] for srow in self.mat]
def __imatmul__(self, multiplier):
self.mat = self @ multiplier
return self.mat
def __rmatmul__(self, multiplicand):
if multiplicand.order[1] != self.order[0]:
raise ValueError("The multiplier was non-conformable under multiplication.")
return [[sum(a*b for a,b in Zip(mrow,scol)) for scol in Zip(*self.mat)] for mrow in multiplicand.mat]
Remarque:
__rmatmul__est requis pour quea @ betb @ afonctionnent correctement;__imatmul__est requis pour quea @= bfonctionne correctement;- Si une matrice est non-conforme à la multiplication, cela signifie qu'elle ne peut pas être multipliée, généralement parce qu'elle a plus ou moins de lignes qu'il n'y a de colonnes dans le multiplicande.
EDIT: Je viens de me rendre compte que __rmatmul__ n’est pas obligatoire, car il n’est utilisé que pour évaluer a @ b si a n’a pas de méthode __matmul__. Puisque je ne multiplie que les matrices par d'autres instances de matrix, a aura aura une méthode __matmul__, mais si je l'édite pour que les opérations fonctionnent, disons, avec des tableaux 2D, je devrai rajouter le __rmatmul__ afin list @ matrix fonctionne aussi bien que matrix @ list.
La faute se produit ici:
C[i][j]+=A[i][k]*B[k][j]
Il se bloque lorsque k = 2. En effet, le Tuple A[i] n'a que 2 valeurs et vous ne pouvez donc l'appeler que par un A [i] [1] avant qu'il ne commette d'erreur.
EDIT: Écoutez la réponse de Gérard aussi, votre C est faux. Ce devrait être C=[[0 for row in range(len(A))] for col in range(len(A[0]))].
Juste un conseil: vous pouvez remplacer la première boucle par une multiplication, ce serait donc C=[[0]*len(A) for col in range(len(A[0]))]
Multiplication de matrices en python pur.
def matmult(m1,m2):
r=[]
m=[]
for i in range(len(m1)):
for j in range(len(m2[0])):
sums=0
for k in range(len(m2)):
sums=sums+(m1[i][k]*m2[k][j])
r.append(sums)
m.append(r)
r=[]
return m
def matrixmult (A, B):
C = [[0 for row in range(len(A))] for col in range(len(B[0]))]
for i in range(len(A)):
for j in range(len(B[0])):
for k in range(len(B)):
C[i][j] += A[i][k]*B[k][j]
return C
à la deuxième ligne, vous devriez changer
C = [[0 for row in range(len(B[0]))] for col in range(len(A))]
La forme de votre matrice C est fausse; c'est la transposition de ce que vous voulez réellement. (Mais je suis d'accord avec ulmangt: la bonne chose est presque certainement d'utiliser numpy, vraiment.)
Bon mot:
def dot(m1, m2):
return [
[sum(x * y for x, y in Zip(m1_r, m2_c)) for m2_c in Zip(*m2)] for m1_r in m1
]
Explication:
Zip (* m2) - obtient une colonne de la deuxième matrice
Zip (m1_r, m2_c) - crée un tuple à partir de la ligne m1 et de la colonne m2
sum (...) - somme multiplication ligne * col
Tester:
m1 = [[1, 2, 3], [4, 5, 6]]
m2 = [[7, 8], [9, 10], [11, 12]]
result = dot(m1, m2)
assert result == [[58, 64], [139, 154]]
Toutes les réponses ci-dessous vous renverraient la liste.Votre besoin de la convertir en matrice
def MATMUL(X, Y):
rows_A = len(X)
cols_A = len(X[0])
rows_B = len(Y)
cols_B = len(Y[0])
if cols_A != rows_B:
print "Matrices are not compatible to Multiply. Check condition C1==R2"
return
# Create the result matrix
# Dimensions would be rows_A x cols_B
C = [[0 for row in range(cols_B)] for col in range(rows_A)]
print C
for i in range(rows_A):
for j in range(cols_B):
for k in range(cols_A):
C[i][j] += A[i][k] * B[k][j]
C = numpy.matrix(C).reshape(len(A),len(B[0]))
return C
m=input("row")
n=input("col")
X=[]
for i in range (m):
m1=[]
for j in range (n):
m1.append(input("num"))
X.append(m1)
Y=[]
for i in range (m):
n1=[]
for j in range (n):
n1.append(input("num"))
Y.append(n1)
# result is 3x3
result = [[0,0,0],
[0,0,0],
[0,0,0]]
# iterate through rows of X
for i in range(len(X)):
# iterate through columns of Y
for j in range(len(Y[0])):
# iterate through rows of Y
for k in range(len(Y)):
result[i][j] += X[i][k] * Y[k][j]
for r in result:
print(r)