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normal Matrix
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sp.Matrix([[1,2],[3,4]])
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Matrix with all zeros
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sp.zeros(4,5)
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Matrix with all ones
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sp.ones(4,5)
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Square matrix with all zeros
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sp.zeros(5)
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Square matrix with all ones
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sp.ones(5)
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Identity matrix
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sp.eyes(5)
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Diagonal Matrix
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sp.diag(1,2,3,4)
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Generate element with func(i,j)
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sp.Matrix(2,3,func)
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Delete the i-th row
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M.row_del(i)
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Delete the j-th column
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M.col_del(j)
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Row join M1 and M2
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M1.row_join(M2)
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Column join M1 and M2
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M1.col_join(M2)
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get the element in M at (i,j)
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M[i,j]
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get the i-th row in M
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M.row(i)
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get the i-th row in M
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M[i,:]
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get the j-th column in M
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M.col(j)
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get the j-th column in M
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M[:,j]
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get the i-th and the k-th rows
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M[[i,k],:]
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get the j-th and the k-th columns
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M[:,[j,k]]
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get rows from i to k
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M[i:k,:]
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get columns from j to k
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M[:,j:k]
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get sub-matrix (row i to k,col j to l)
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M[i:k,j:l]
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Sum
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A+B
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Substraction
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A-B
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Matrix Multiply
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A*B
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Scalar Multiply
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5*A
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Elementwise product
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sp.matrix_multiply_elementwise(A,B)
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Transpose
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A.T
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Determinant
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A.det()
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Inverse
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A.inv()
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Condition Number
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A.condition_number()
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Row count
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A.rows
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Column count
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A.cols
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Trace
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A.trace()
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Replacement
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m.row_op(i, lambda ele,col:ele+m.row(j)[col]*c)
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Interchange
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M.row_swap(i,j)
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Scaling
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m.row_op(i,lambda ele,col:ele*c)
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Echelon From
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M.echelon_form()
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Reduced Echelon Form
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M.rref()
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Solve AX=B (B can be a matrix)
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x,freevars=A.gauss_jordan_solve(B)
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least-square fit Ax=b
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A.solve_least_squares(b)
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solve Ax=b
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A.solve(b)
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Basis of column space
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M.columnspace()
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Basis of null space
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M.nullspace()
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Basis of row space
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M.rowspace()
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Rank
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M.rank()
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Find the eigenvalues
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M.eigenvals()
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Find the eignevalues and the corresponding eigenspace
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M.eigenvects()
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Diagonalize a matrix
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P, D = M.diagonalize()
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test if the matrix is diagonalizable
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M.is_diagonalizable
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Calculate Jordan From
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P, J = M.jordan_form()
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LU Decomposition(PA=LU)
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P,L,U=A.LUdecomposition()
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QR Decomposition
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Q,R=A.QRdecomposition()
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Create a column vector
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v=sp.Matrix([1,2,3])
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dot product
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v1.dot(v2)
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cross product
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v1.cross(v2)
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length of the vector
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v.norm()
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normalize of vector
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v.normalize()
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the projection of v1 on v2
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v1.project(v2)
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Gram-Schmidt orthogonalize
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sp.GramSchmidt([v1,v2,v3])
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Gram-Schmidt orthogonalize with normalization
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sp.GramSchmidt([v1,v2,v3],True)
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Singular values
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M.singlular_values()
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Create a matrix by block
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M=sp.Matrix([[A,B],[C,D]])
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