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Linear Algebra Using SymPy Cheat Sheet by

Import SymPy

import sympy as sp

Matrix Creation

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)

Matrix Modifi­cation

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)

Indexi­ng(­Sli­cing)

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]
Note: All indices start from 0
 

Basic opertaions

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()

Elementary Row Operations

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)

Linear Equations

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)

Vector Space

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

Eigenv­alues amd Eigenv­ectors

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()

Decomp­osition

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

Vector Operations

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()

Block Matrix

Create a matrix by block
M=sp.M­atr­ix(­[[A­,B]­,[C­,D]])

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