Tutorial Matriks dengan Python Numpy (Operasi Baris Elementer)

Assalaamu'alaikum, Sahabat.
Python itu canggih ternyata. Selain cepat, data besar juga mumpuni lho. Nah, kali ini aku ingin nge share pengalamanku tentang matriks. Paket tambahan yang kita perlukan hanyalah numpy. Ya. Itu saja. Kalau kita pecinta ubuntu, dan ada koneksi internet, cukup kita ketikkan di terminal:
sudo apt-get install python-numpy
Lha kalau kita masih pakai windows, ya cukup download dan instal file ini saja: http://sourceforge.net/projects/numpy/files/. Langsung saja, yuk kita pelajari di terminalku ini, Sahabat.. :)
muktyas@muktyas-Aspire-4830Z:~$ python Python 2.7.10 (default, Oct 14 2015, 16:09:02) [GCC 5.2.1 20151010] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> import numpy as np >>> np.ones(4) array([ 1., 1., 1., 1.]) >>> np.eye(4) array([[ 1., 0., 0., 0.], [ 0., 1., 0., 0.], [ 0., 0., 1., 0.], [ 0., 0., 0., 1.]]) >>> A=[[1,2],[3,4]] >>> B=np.array(A) >>> B array([[1, 2], [3, 4]]) >>> c=[[4,1],[2,5]] >>> C=np.array(c) >>> B array([[1, 2], [3, 4]]) >>> C array([[4, 1], [2, 5]]) >>> B*C array([[ 4, 2], [ 6, 20]]) >>> np.mat(B) matrix([[1, 2], [3, 4]]) >>> np.mat(B),np.mat(C) (matrix([[1, 2], [3, 4]]), matrix([[4, 1], [2, 5]])) >>> np.mat(B)*np.mat(C) matrix([[ 8, 11], [20, 23]]) >>> np.zeros(3,2) Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: data type not understood >>> np.zeros(3) array([ 0., 0., 0.]) >>> np.zeros((3,2)) array([[ 0., 0.], [ 0., 0.], [ 0., 0.]]) >>> mB, mC = np.mat(B),np.matrix(C) >>> mB, mC (matrix([[1, 2], [3, 4]]), matrix([[4, 1], [2, 5]])) >>> mB+mC matrix([[5, 3], [5, 9]]) >>> mB mC File "<stdin>", line 1 mB mC ^ SyntaxError: invalid syntax >>> print mB mC File "<stdin>", line 1 print mB mC ^ SyntaxError: invalid syntax >>> print mB+" " + mC Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: unsupported operand type(s) for +: 'matrix' and 'str' >>> print str(mB)+" " + str(mC) [[1 2] [3 4]] [[4 1] [2 5]] >>> mB matrix([[1, 2], [3, 4]]) >>> mC matrix([[4, 1], [2, 5]]) >>> mB+mC matrix([[5, 3], [5, 9]]) >>> mB-mC matrix([[-3, 1], [ 1, -1]]) >>> mB*mC matrix([[ 8, 11], [20, 23]]) >>> mB/mC matrix([[0, 2], [1, 0]]) >>> mB/mC*mC matrix([[ 4, 10], [ 4, 1]]) >>> mC*mB/mC matrix([[ 1, 12], [ 8, 4]]) >>> mC*(mB/mC) matrix([[1, 8], [5, 4]]) >>> (mC*mB)/mC matrix([[ 1, 12], [ 8, 4]]) >>> (mB/mC)*mC matrix([[ 4, 10], [ 4, 1]]) >>> mB matrix([[1, 2], [3, 4]]) >>> mB(1) Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: 'matrix' object is not callable >>> mB[1] matrix([[3, 4]]) >>> mB[0] matrix([[1, 2]]) >>> mB[0,1] 2 >>> mB matrix([[1, 2], [3, 4]]) >>> mB[1]-3*mB[0] matrix([[ 0, -2]]) >>> mD=mB >>> mB[1]=mB[1]-3*mB[0] >>> mB matrix([[ 1, 2], [ 0, -2]]) >>> >>> A=np.mat([[1,2.],[3,4]]) >>> A matrix([[ 1., 2.], [ 3., 4.]]) >>> np.mat.eye(2) Traceback (most recent call last): File "<stdin>", line 1, in <module> AttributeError: 'function' object has no attribute 'eye' >>> np.eye(2) array([[ 1., 0.], [ 0., 1.]]) >>> np.mat(np.eye(2)) matrix([[ 1., 0.], [ 0., 1.]]) >>> I=np.mat(np.eye(2)) >>> np.hstack(A,I) Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: hstack() takes exactly 1 argument (2 given) >>> np.hstack([A,I]) matrix([[ 1., 2., 1., 0.], [ 3., 4., 0., 1.]]) >>> J=np.eye(2) >>> np.hstack([A,J]) matrix([[ 1., 2., 1., 0.], [ 3., 4., 0., 1.]]) >>> Aug=np.hstack([A,J]) >>> Aug matrix([[ 1., 2., 1., 0.], [ 3., 4., 0., 1.]]) >>> Aug[[0,1]]=Aug[[0,1]] >>> Aug matrix([[ 1., 2., 1., 0.], [ 3., 4., 0., 1.]]) >>> Aug[[0,1]]=Aug[[1,0]] >>> Aug matrix([[ 3., 4., 0., 1.], [ 1., 2., 1., 0.]]) >>>

Ini juga bisa:
muktyas@muktyas-Aspire-4830Z:~$ python Python 2.7.10 (default, Oct 14 2015, 16:09:02) [GCC 5.2.1 20151010] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> A Traceback (most recent call last): File "<stdin>", line 1, in <module> NameError: name 'A' is not defined >>> import numpy as np >>> A=np.mat([[3,1,2],[0,1,-5.],[1,1,-1]]) >>> A matrix([[ 3., 1., 2.], [ 0., 1., -5.], [ 1., 1., -1.]]) >>> I=np.eyes(3) Traceback (most recent call last): File "<stdin>", line 1, in <module> AttributeError: 'module' object has no attribute 'eyes' >>> I=np.eye(3) >>> I array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]]) >>> AI=np.hstack([A,I]) >>> AI matrix([[ 3., 1., 2., 1., 0., 0.], [ 0., 1., -5., 0., 1., 0.], [ 1., 1., -1., 0., 0., 1.]]) >>> B=AI >>> B matrix([[ 3., 1., 2., 1., 0., 0.], [ 0., 1., -5., 0., 1., 0.], [ 1., 1., -1., 0., 0., 1.]]) >>> B[[0,2],:] = B[[2,0],:] >>> B matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., -5., 0., 1., 0.], [ 3., 1., 2., 1., 0., 0.]]) >>> B[:,[1,2]] = B[:,[2,1]] >>> B matrix([[ 1., -1., 1., 0., 0., 1.], [ 0., -5., 1., 0., 1., 0.], [ 3., 2., 1., 1., 0., 0.]]) >>> B[[1,2]]=B[[2,1]] >>> B matrix([[ 1., -1., 1., 0., 0., 1.], [ 3., 2., 1., 1., 0., 0.], [ 0., -5., 1., 0., 1., 0.]]) >>> def gb(M,dari,ke): ... M[[dari,ke],:] = M[[ke,dari],:] ... >>> def gk(M,dari,ke): ... M[:,[dari,ke]] = M[:,[ke,dari]] ... >>> AI matrix([[ 1., -1., 1., 0., 0., 1.], [ 3., 2., 1., 1., 0., 0.], [ 0., -5., 1., 0., 1., 0.]]) >>> gb(AI,0,2) >>> AI matrix([[ 0., -5., 1., 0., 1., 0.], [ 3., 2., 1., 1., 0., 0.], [ 1., -1., 1., 0., 0., 1.]]) >>> gk(AI,2,3) >>> AI matrix([[ 0., -5., 0., 1., 1., 0.], [ 3., 2., 1., 1., 0., 0.], [ 1., -1., 0., 1., 0., 1.]]) >>> A matrix([[ 3., 1., 2.], [ 0., 1., -5.], [ 1., 1., -1.]]) >>> AI=np.hstack([A,I]) >>> AI matrix([[ 3., 1., 2., 1., 0., 0.], [ 0., 1., -5., 0., 1., 0.], [ 1., 1., -1., 0., 0., 1.]]) >>> sekarang lakukan obe untuk mendapatkan inversnya File "<stdin>", line 1 sekarang lakukan obe untuk mendapatkan inversnya ^ SyntaxError: invalid syntax >>> >>>>>>>>>>>>>>>>>>>>>>>> File "<stdin>", line 1 >>>>>>>>>>>>>>>>>>>>>>>> ^ SyntaxError: invalid syntax >>> gb(AI,0,2) >>> AI matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., -5., 0., 1., 0.], [ 3., 1., 2., 1., 0., 0.]]) >>> AI[2] matrix([[ 3., 1., 2., 1., 0., 0.]]) >>> AI[2]=AI[2]-3*AI[1] >>> AI matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., -5., 0., 1., 0.], [ 3., -2., 17., 1., -3., 0.]]) >>> AI[2]=AI[2]+3*AI[1] >>> AI matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., -5., 0., 1., 0.], [ 3., 1., 2., 1., 0., 0.]]) >>> AI[2]=AI[2]-3*AI[0] >>> AI matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., -5., 0., 1., 0.], [ 0., -2., 5., 1., 0., -3.]]) >>> AI[2]=AI[2]-(-2)*AI[1] >>> AI matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., -5., 0., 1., 0.], [ 0., 0., -5., 1., 2., -3.]]) >>> def ob(M,b1,b2,pengali): ... M[b2]=M[b2]-pengali*M[b1] ... >>> ob(AI,2,1,1) >>> AI matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., 0., -1., -1., 3.], [ 0., 0., -5., 1., 2., -3.]]) >>> ob(AI,2,1,-1) >>> AI matrix([[ 1., 1., -1., 0., 0., 1.], [ 0., 1., -5., 0., 1., 0.], [ 0., 0., -5., 1., 2., -3.]]) >>> ob(AI,1,0,1) >>> AI matrix([[ 1., 0., 4., 0., -1., 1.], [ 0., 1., -5., 0., 1., 0.], [ 0., 0., -5., 1., 2., -3.]]) >>> AI[2]=AI[2]/5 >>> AI matrix([[ 1. , 0. , 4. , 0. , -1. , 1. ], [ 0. , 1. , -5. , 0. , 1. , 0. ], [ 0. , 0. , -1. , 0.2, 0.4, -0.6]]) >>> AI[2]=AI[2]*-1 >>> AI matrix([[ 1. , 0. , 4. , 0. , -1. , 1. ], [ 0. , 1. , -5. , 0. , 1. , 0. ], [ 0. , 0. , 1. , -0.2, -0.4, 0.6]]) >>> ob(AI,2,0,4) >>> AI matrix([[ 1. , 0. , 0. , 0.8, 0.6, -1.4], [ 0. , 1. , -5. , 0. , 1. , 0. ], [ 0. , 0. , 1. , -0.2, -0.4, 0.6]]) >>> ob(AI,2,1,-5) >>> AI matrix([[ 1. , 0. , 0. , 0.8, 0.6, -1.4], [ 0. , 1. , 0. , -1. , -1. , 3. ], [ 0. , 0. , 1. , -0.2, -0.4, 0.6]]) >>> inversA=AI[:,[3,5]] >>> inversA matrix([[ 0.8, -1.4], [-1. , 3. ], [-0.2, 0.6]]) >>> inversA=AI[:,[3:5]] File "<stdin>", line 1 inversA=AI[:,[3:5]] ^ SyntaxError: invalid syntax >>> AI[:,[3,5]] matrix([[ 0.8, -1.4], [-1. , 3. ], [-0.2, 0.6]]) >>> AI[:,[3,4,5]] matrix([[ 0.8, 0.6, -1.4], [-1. , -1. , 3. ], [-0.2, -0.4, 0.6]]) >>> AI[:,[{3,5}]] Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/usr/lib/python2.7/dist-packages/numpy/matrixlib/defmatrix.py", line 316, in __getitem__ out = N.ndarray.__getitem__(self, index) TypeError: long() argument must be a string or a number, not 'set' >>> AI[:,[[3,5]]] matrix([[ 8.00000000e-001], [ 3.47192463e-310], [ 5.29773656e-315]]) >>> AI[:,[(3,5)]] matrix([[ 8.00000000e-001], [ 3.47192463e-310], [ 5.29773656e-315]]) >>> AI[:,[3..5]] File "<stdin>", line 1 AI[:,[3..5]] ^ SyntaxError: invalid syntax >>> AI[:,[{3..5}]] File "<stdin>", line 1 AI[:,[{3..5}]] ^ SyntaxError: invalid syntax >>> AI[:,[3,4,5]] matrix([[ 0.8, 0.6, -1.4], [-1. , -1. , 3. ], [-0.2, -0.4, 0.6]]) >>> AI[:,[3:5]] File "<stdin>", line 1 AI[:,[3:5]] ^ SyntaxError: invalid syntax >>> AI[:,[[3:5]]] File "<stdin>", line 1 AI[:,[[3:5]]] ^ SyntaxError: invalid syntax >>> AI[3:5] matrix([], shape=(0, 6), dtype=float64) >>> AI[[3:5]] File "<stdin>", line 1 AI[[3:5]] ^ SyntaxError: invalid syntax >>> AI matrix([[ 1. , 0. , 0. , 0.8, 0.6, -1.4], [ 0. , 1. , 0. , -1. , -1. , 3. ], [ 0. , 0. , 1. , -0.2, -0.4, 0.6]]) >>> AI(2) Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: 'matrix' object is not callable >>> AI[2] matrix([[ 0. , 0. , 1. , -0.2, -0.4, 0.6]]) >>> AI[1:2] matrix([[ 0., 1., 0., -1., -1., 3.]]) >>> AI[1:3] matrix([[ 0. , 1. , 0. , -1. , -1. , 3. ], [ 0. , 0. , 1. , -0.2, -0.4, 0.6]]) >>> AI[:,[3:6]] File "<stdin>", line 1 AI[:,[3:6]] ^ SyntaxError: invalid syntax >>> AI[[3:6]] File "<stdin>", line 1 AI[[3:6]] ^ SyntaxError: invalid syntax >>> AI.T[[3:6]] File "<stdin>", line 1 AI.T[[3:6]] ^ SyntaxError: invalid syntax >>> AI.T matrix([[ 1. , 0. , 0. ], [ 0. , 1. , 0. ], [ 0. , 0. , 1. ], [ 0.8, -1. , -0.2], [ 0.6, -1. , -0.4], [-1.4, 3. , 0.6]]) >>> AI matrix([[ 1. , 0. , 0. , 0.8, 0.6, -1.4], [ 0. , 1. , 0. , -1. , -1. , 3. ], [ 0. , 0. , 1. , -0.2, -0.4, 0.6]]) >>> AI[:,[3,4,5]] matrix([[ 0.8, 0.6, -1.4], [-1. , -1. , 3. ], [-0.2, -0.4, 0.6]]) >>> id=range(3,6) >>> id [3, 4, 5] >>> AI[:,range(3,6)] matrix([[ 0.8, 0.6, -1.4], [-1. , -1. , 3. ], [-0.2, -0.4, 0.6]]) >>> >>> AI[:,3:] matrix([[ 0.8, 0.6, -1.4], [-1. , -1. , 3. ], [-0.2, -0.4, 0.6]]) >>> AI[:,2:] matrix([[ 0. , 0.8, 0.6, -1.4], [ 0. , -1. , -1. , 3. ], [ 1. , -0.2, -0.4, 0.6]]) >>> AI[:,2:4] matrix([[ 0. , 0.8], [ 0. , -1. ], [ 1. , -0.2]]) >>> AI[:,2:5] matrix([[ 0. , 0.8, 0.6], [ 0. , -1. , -1. ], [ 1. , -0.2, -0.4]]) >>> 2:5 File "<stdin>", line 1 2:5 ^ SyntaxError: invalid syntax >>> [2:5] File "<stdin>", line 1 [2:5] ^ SyntaxError: invalid syntax >>>

Aku juga sempat ngoprek dan mbuat sebuah file yang isinya fungsi-fungsi dari OBE. Ada tiga kalau ndak salah, yaitu tukar baris, tambahkan baris dengan berapa kalinya baris lain, dan kalikan baris dengan suatu nilai. Aku mbuat file obe.py kayak gini:
#dibuat oleh muktyas #operasi baris elementer #obe import numpy as np #tukar baris def tb(Matriks,b0,b1): Matriks[[b0,b1]] = Matriks[[b1,b0]] #tukar kolom def tk(Matriks,k0,k1): Matriks[:,[k0,k1]] = Matriks[:,[k1,k0]] #operasi baris #bt = bt - n*ba #bt: baris tujuan, ba: baris awal def ob(Matriks,bt,ba,n): Matriks[bt] = Matriks[bt] - n*Matriks[ba] #kali baris dengan suatu konstanta def kali(Matriks,baris,n): Matriks[baris] = n*Matriks[baris]

Okay. Selanjutnya ini contoh penggunaannya, Sahabat. Kita buka terminal di lokasi yang sama dengan kita nyimpan obe.py itu. Lihat lagi, ya. muktyas@muktyas-Aspire-4830Z:~/Dropbox/lain$ python Python 2.7.10 (default, Oct 14 2015, 16:09:02) [GCC 5.2.1 20151010] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> from obe import * >>> A=np.arange(4*5).reshape(4,5) >>> A array([[ 0, 1, 2, 3, 4], [ 5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19]]) >>> tb(A,1,2) >>> A array([[ 0, 1, 2, 3, 4], [10, 11, 12, 13, 14], [ 5, 6, 7, 8, 9], [15, 16, 17, 18, 19]]) >>> tb(A,0,2) >>> A array([[ 5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [ 0, 1, 2, 3, 4], [15, 16, 17, 18, 19]]) >>> ob(A,1,0,2) >>> A array([[ 5, 6, 7, 8, 9], [ 0, -1, -2, -3, -4], [ 0, 1, 2, 3, 4], [15, 16, 17, 18, 19]]) >>> ob(A,3,0,3) >>> A array([[ 5, 6, 7, 8, 9], [ 0, -1, -2, -3, -4], [ 0, 1, 2, 3, 4], [ 0, -2, -4, -6, -8]]) >>> kali(A,1,-1) >>> A array([[ 5, 6, 7, 8, 9], [ 0, 1, 2, 3, 4], [ 0, 1, 2, 3, 4], [ 0, -2, -4, -6, -8]]) >>> kali(A,3,9) >>> A array([[ 5, 6, 7, 8, 9], [ 0, 1, 2, 3, 4], [ 0, 1, 2, 3, 4], [ 0, -18, -36, -54, -72]]) >>>

Update:
Sahabat, aku mbuat updatetan dari file obe.py biar lebih manusiawi. Maksudnya baris pertama dari matriks M ya nanti kita masukkannya tetep 1, bukan nol. Kayak gini file obe2.py ku:
#dibuat oleh muktyas #operasi baris elementer #obe2 import numpy as np #dimulai dari 1, bukan nol. #tukar baris def tb(Matriks,b0,b1): Matriks[[b0-1,b1-1]] = Matriks[[b1-1,b0-1]] #tukar kolom def tk(Matriks,k0,k1): Matriks[:,[k0-1,k1-1]] = Matriks[:,[k1-1,k0-1]] #operasi baris #bt = bt + n*ba #bt: baris tujuan, ba: baris awal def ob(Matriks,bt,ba,n): Matriks[bt-1] = Matriks[bt-1] + n*Matriks[ba-1] #kali baris dengan suatu konstanta def kali(Matriks,baris,n): Matriks[baris-1] = n*Matriks[baris-1]
Dengan pakai file obe2.py, kalau mau melakukan operasi baris 3 + 2 * baris 1 pada matriks M dengan kode seperti ini.
ob(M,3,1,2)

Nah, begitu.
Belajar dengan langsung praktek itu enak kok.
Semoga bermanfaat, Sahabat.
Kalau kotak kodenya kekecilan, bisa diubah ukurannya. Klik dan drag di pojok kanan bawah aja.
Assalaamu'alaikum wa rahmatullaah wa barakaatuh.