Linear Algebra Using SymPy Cheat Sheet by royqh1979

normal Matrix	sp.Matrix([[1,2­],[­3,4]])
Matrix with all zeros	sp.zeros(4,5)
Matrix with all ones	sp.ones(4,5)
Square matrix with all zeros	sp.zeros(5)
Square matrix with all ones	sp.ones(5)
Identity matrix	sp.eyes(5)
Diagonal Matrix	sp.diag(1,2,3,4)
Generate element with func(i,j)	sp.Matrix(2,3,func)

Delete the i-th row	M.row_­del(i)
Delete the j-th column	M.col_­del(j)
Row join M1 and M2	M1.row­_jo­in(M2)
Column join M1 and M2	M1.col­_jo­in(M2)

get the element in M at (i,j)	M[i,j]
get the i-th row in M	M.row(i)
get the i-th row in M	M[i,:]
get the j-th column in M	M.col(j)
get the j-th column in M	M[:,j]
get the i-th and the k-th rows	M[[i,k],:]
get the j-th and the k-th columns	M[:,[j,k]]
get rows from i to k	M[i:k,:]
get columns from j to k	M[:,j:k]
get sub-matrix (row i to k,col j to l)	M[i:k,j:l]

Sum	A+B
Substr­action	A-B
Matrix Multiply	A*B
Scalar Multiply	5*A
Elemen­twise product	sp.mat­rix­_mu­lti­ply­_el­eme­ntw­ise­(A,B)
Transpose	A.T
Determ­inant	A.det()
Inverse	A.inv()
Condition Number	A.cond­iti­on_­num­ber()
Row count	A.rows
Column count	A.cols
Trace	A.trace()

Replac­ement	m.row_­op(i, lambda ele,co­l:e­le+­m.r­ow(­j)[­col]*c)
Interc­hange	M.row_­swa­p(i,j)
Scaling	m.row_­op(i,lambda ele,co­l:e­le*c)

Echelon From	M.eche­lon­_form()
Reduced Echelon Form	M.rref()
Solve AX=B (B can be a matrix)	x,free­var­s=A.ga­uss­_jo­rda­n_s­olve(B)
least-­square fit Ax=b	A.solv­e_l­eas­t_s­qua­res(b)
solve Ax=b	A.solve(b)

Basis of column space	M.colu­mns­pace()
Basis of null space	M.null­space()
Basis of row space	M.rows­pace()
Rank	M.rank()

Find the eigenv­alues	M.eige­nvals()
Find the eignev­alues and the corres­ponding eigenspace	M.eige­nve­cts()
Diagon­alize a matrix	P, D = M.diag­ona­lize()
test if the matrix is diagon­ali­zable	M.is_d­iag­ona­lizable
Calculate Jordan From	P, J = M.jord­an_­form()

LU Decomp­osi­tio­n(P­A=LU)	P,L,U=­A.L­Ude­com­pos­ition()
QR Decomp­osition	Q,R=A.Q­Rd­eco­mpo­sit­ion()

Create a column vector	v=sp.M­atr­ix(­[1,­2,3])
dot product	v1.dot(v2)
cross product	v1.cro­ss(v2)
length of the vector	v.norm()
normalize of vector	v.norm­alize()
the projection of v1 on v2	v1.pro­jec­t(v2)
Gram-S­chmidt orthog­onalize	sp.Gra­mSc­hmi­dt(­[v1­,v2­,v3])
Gram-S­chmidt orthog­onalize with normal­ization	sp.Gra­mSc­hmi­dt(­[v1­,v2­,v3­],True)
Singular values	M.sing­lul­ar_­val­ues()

Create a matrix by block

M=sp.M­atr­ix(­[[A­,B]­,[C­,D]])