Matrices (linear algebra)¶
Creating Matrices¶
The linear algebra module is designed to be as simple as possible. First, we
import and declare our first Matrix
object:
>>> init_printing(pretty_print=True, use_unicode=False, wrap_line=False, no_global=True)
>>> M = Matrix([[1, 0, 0], [0, 0, 0]]); M
[1 0 0]
[ ]
[0 0 0]
>>> Matrix([M, (0, 0, 1)])
[1 0 0 ]
[ ]
[0 0 0 ]
[ ]
[0 0 1]
>>> Matrix([[1, 2, 3]])
[1 2 3]
>>> Matrix([1, 2, 3])
[1]
[ ]
[2]
[ ]
[3]
In addition to creating a matrix from a list of appropriatelysized lists and/or matrices, Diofant also supports more advanced methods of matrix creation including a single list of values and dimension inputs:
>>> Matrix(2, 3, [1, 2, 3, 4, 5, 6])
[1 2 3]
[ ]
[4 5 6]
More interesting (and useful), is the ability to use a 2variable function
(or lambda
) to create a matrix. Here we create an indicator function which
is 1 on the diagonal and then use it to make the identity matrix:
>>> def f(i, j):
... if i == j:
... return 1
... else:
... return 0
...
>>> Matrix(4, 4, f)
[1 0 0 0]
[ ]
[0 1 0 0]
[ ]
[0 0 1 0]
[ ]
[0 0 0 1]
Finally let’s use lambda
to create a 1line matrix with 1’s in the even
permutation entries:
>>> Matrix(3, 4, lambda i, j: 1  (i + j)%2)
[1 0 1 0]
[ ]
[0 1 0 1]
[ ]
[1 0 1 0]
There are also a couple of special constructors for quick matrix construction:
eye
is the identity matrix, zeros
and ones
for matrices of all
zeros and ones, respectively, and diag
to put matrices or elements along
the diagonal:
>>> eye(4)
[1 0 0 0]
[ ]
[0 1 0 0]
[ ]
[0 0 1 0]
[ ]
[0 0 0 1]
>>> zeros(2)
[0 0]
[ ]
[0 0]
>>> zeros(2, 5)
[0 0 0 0 0]
[ ]
[0 0 0 0 0]
>>> ones(3)
[1 1 1]
[ ]
[1 1 1]
[ ]
[1 1 1]
>>> ones(1, 3)
[1 1 1]
>>> diag(1, Matrix([[1, 2], [3, 4]]))
[1 0 0]
[ ]
[0 1 2]
[ ]
[0 3 4]
Basic Manipulation¶
While learning to work with matrices, let’s choose one where the entries are readily identifiable. One useful thing to know is that while matrices are 2dimensional, the storage is not and so it is allowable  though one should be careful  to access the entries as if they were a 1d list.
>>> M = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
>>> M[4]
5
Now, the more standard entry access is a pair of indices which will always return the value at the corresponding row and column of the matrix:
>>> M[1, 2]
6
>>> M[0, 0]
1
>>> M[1, 1]
5
Since this is Python we’re also able to slice submatrices; slices always give a matrix in return, even if the dimension is 1 x 1:
>>> M[0:2, 0:2]
[1 2]
[ ]
[4 5]
>>> M[2:2, 2]
[]
>>> M[:, 2]
[3]
[ ]
[6]
>>> M[:1, 2]
[3]
In the second example above notice that the slice 2:2 gives an empty range. Note also (in keeping with 0based indexing of Python) the first row/column is 0.
You cannot access rows or columns that are not present unless they are in a slice:
>>> M[:, 10] # the 10th column (not there)
Traceback (most recent call last):
...
IndexError: Index out of range: a[[0, 10]]
>>> M[:, 10:11] # the 10th column (if there)
[]
>>> M[:, :10] # all columns up to the 10th
[1 2 3]
[ ]
[4 5 6]
Slicing an empty matrix works as long as you use a slice for the coordinate that has no size:
>>> Matrix(0, 3, [])[:, 1]
[]
Slicing gives a copy of what is sliced, so modifications of one object do not affect the other:
>>> M2 = M[:, :]
>>> M2[0, 0] = 100
>>> M[0, 0] == 100
False
Notice that changing M2
didn’t change M
. Since we can slice, we can also assign
entries:
>>> M = Matrix(([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]))
>>> M
[1 2 3 4 ]
[ ]
[5 6 7 8 ]
[ ]
[9 10 11 12]
[ ]
[13 14 15 16]
>>> M[2, 2] = M[0, 3] = 0
>>> M
[1 2 3 0 ]
[ ]
[5 6 7 8 ]
[ ]
[9 10 0 12]
[ ]
[13 14 15 16]
as well as assign slices:
>>> M = Matrix(([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]))
>>> M[2:, 2:] = Matrix(2, 2, lambda i, j: 0)
>>> M
[1 2 3 4]
[ ]
[5 6 7 8]
[ ]
[9 10 0 0]
[ ]
[13 14 0 0]
All the standard arithmetic operations are supported:
>>> M = Matrix(([1, 2, 3], [4, 5, 6], [7, 8, 9]))
>>> M  M
[0 0 0]
[ ]
[0 0 0]
[ ]
[0 0 0]
>>> M + M
[2 4 6 ]
[ ]
[8 10 12]
[ ]
[14 16 18]
>>> M * M
[30 36 42 ]
[ ]
[66 81 96 ]
[ ]
[102 126 150]
>>> M2 = Matrix(3, 1, [1, 5, 0])
>>> M*M2
[11]
[ ]
[29]
[ ]
[47]
>>> M**2
[30 36 42 ]
[ ]
[66 81 96 ]
[ ]
[102 126 150]
As well as some useful vector operations:
>>> del M[0, :]
>>> M
[4 5 6]
[ ]
[7 8 9]
>>> del M[:, 1]
>>> M
[4 6]
[ ]
[7 9]
>>> v1 = Matrix([1, 2, 3])
>>> v2 = Matrix([4, 5, 6])
>>> v3 = v1.cross(v2)
>>> v1.dot(v2)
32
>>> v2.dot(v3)
0
>>> v1.dot(v3)
0
We can also ‘’glue’’ together matrices of the appropriate size:
>>> M1 = eye(3)
>>> M2 = zeros(3, 4)
>>> M1.row_join(M2)
[1 0 0 0 0 0 0]
[ ]
[0 1 0 0 0 0 0]
[ ]
[0 0 1 0 0 0 0]
>>> M3 = zeros(4, 3)
>>> M1.col_join(M3)
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
[ ]
[0 0 0]
[ ]
[0 0 0]
[ ]
[0 0 0]
[ ]
[0 0 0]
Operations on entries¶
We are not restricted to having multiplication between two matrices:
>>> M = eye(3)
>>> 2*M
[2 0 0]
[ ]
[0 2 0]
[ ]
[0 0 2]
>>> 3*M
[3 0 0]
[ ]
[0 3 0]
[ ]
[0 0 3]
but we can also apply functions to our matrix entries using applyfunc()
. Here we’ll declare a function that double any input number. Then we apply it to the 3x3 identity matrix:
>>> f = lambda x: 2*x
>>> eye(3).applyfunc(f)
[2 0 0]
[ ]
[0 2 0]
[ ]
[0 0 2]
One more useful matrixwide entry application function is the substitution function. Let’s declare a matrix with symbolic entries then substitute a value. Remember we can substitute anything  even another symbol!:
>>> M = eye(3) * x
>>> M
[x 0 0]
[ ]
[0 x 0]
[ ]
[0 0 x]
>>> M.subs({x: 4})
[4 0 0]
[ ]
[0 4 0]
[ ]
[0 0 4]
>>> y = Symbol('y')
>>> M.subs({x: y})
[y 0 0]
[ ]
[0 y 0]
[ ]
[0 0 y]
Linear algebra¶
Now that we have the basics out of the way, let’s see what we can do with the actual matrices. Of course, one of the first things that comes to mind is the determinant:
>>> M = Matrix(([1, 2, 3], [3, 6, 2], [2, 0, 1]))
>>> M.det()
28
>>> M2 = eye(3)
>>> M2.det()
1
>>> M3 = Matrix(([1, 0, 0], [1, 0, 0], [1, 0, 0]))
>>> M3.det()
0
Another common operation is the inverse: In Diofant, this is computed by Gaussian elimination by default (for dense matrices) but we can specify it be done by \(LU\) decomposition as well:
>>> M2.inv()
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
>>> M2.inv(method="LU")
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
>>> M.inv(method="LU")
[3/14 1/14 1/2 ]
[ ]
[1/28 5/28 1/4]
[ ]
[ 3/7 1/7 0 ]
>>> M * M.inv(method="LU")
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
We can perform a \(QR\) factorization which is handy for solving systems:
>>> A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
>>> Q, R = A.QRdecomposition()
>>> Q
[ ___ ___ ___ ]
[\/ 6 \/ 3 \/ 2 ]
[  ]
[ 6 3 2 ]
[ ]
[ ___ ___ ___ ]
[\/ 6 \/ 3 \/ 2 ]
[   ]
[ 6 3 2 ]
[ ]
[ ___ ___ ]
[\/ 6 \/ 3 ]
[  0 ]
[ 3 3 ]
>>> R
[ ___ ]
[ ___ 4*\/ 6 ___]
[\/ 6  2*\/ 6 ]
[ 3 ]
[ ]
[ ___ ]
[ \/ 3 ]
[ 0  0 ]
[ 3 ]
[ ]
[ ___ ]
[ 0 0 \/ 2 ]
>>> Q*R
[1 1 1]
[ ]
[1 1 3]
[ ]
[2 3 4]
In addition to the solvers in the solver.py
file, we can solve the system Ax=b
by passing the b vector to the matrix A’s LUsolve function. Here we’ll cheat a
little choose A and x then multiply to get b. Then we can solve for x and check
that it’s correct:
>>> A = Matrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
>>> x = Matrix(3, 1, [3, 7, 5])
>>> b = A*x
>>> soln = A.LUsolve(b)
>>> soln
[3]
[ ]
[7]
[ ]
[5]
There’s also a nice GramSchmidt orthogonalizer which will take a set of
vectors and orthogonalize them with respect to another. There is an
optional argument which specifies whether or not the output should also be
normalized, it defaults to False
. Let’s take some vectors and orthogonalize
them  one normalized and one not:
>>> L = [Matrix([2, 3, 5]), Matrix([3, 6, 2]), Matrix([8, 3, 6])]
>>> out1 = GramSchmidt(L)
>>> out2 = GramSchmidt(L, True)
Let’s take a look at the vectors:
>>> for i in out1:
... print(i)
...
Matrix([
[2],
[3],
[5]])
Matrix([
[ 23/19],
[ 63/19],
[47/19]])
Matrix([
[ 1692/353],
[1551/706],
[ 423/706]])
>>> for i in out2:
... print(i)
...
Matrix([
[ sqrt(38)/19],
[3*sqrt(38)/38],
[5*sqrt(38)/38]])
Matrix([
[ 23*sqrt(6707)/6707],
[ 63*sqrt(6707)/6707],
[47*sqrt(6707)/6707]])
Matrix([
[ 12*sqrt(706)/353],
[11*sqrt(706)/706],
[ 3*sqrt(706)/706]])
We can spotcheck their orthogonality with dot() and their normality with norm():
>>> out1[0].dot(out1[1])
0
>>> out1[0].dot(out1[2])
0
>>> out1[1].dot(out1[2])
0
>>> out2[0].norm()
1
>>> out2[1].norm()
1
>>> out2[2].norm()
1
So there is quite a bit that can be done with the module including eigenvalues,
eigenvectors, nullspace calculation, cofactor expansion tools, and so on. From
here one might want to look over the matrices.py
file for all functionality.
MatrixBase Class Reference¶

class
diofant.matrices.matrices.
MatrixBase
[source]¶ 
C
¶ Byelement conjugation.

D
¶ Return Dirac conjugate (if self.rows == 4).
Examples
>>> m = Matrix((0, 1 + I, 2, 3)) >>> m.D Matrix([[0, 1  I, 2, 3]]) >>> m = (eye(4) + I*eye(4)) >>> m[0, 3] = 2 >>> m.D Matrix([ [1  I, 0, 0, 0], [ 0, 1  I, 0, 0], [ 0, 0, 1 + I, 0], [ 2, 0, 0, 1 + I]])
If the matrix does not have 4 rows an AttributeError will be raised because this property is only defined for matrices with 4 rows.
>>> Matrix(eye(2)).D Traceback (most recent call last): ... AttributeError: Matrix has no attribute D.

H
¶ Return Hermite conjugate.
Examples
>>> m = Matrix((0, 1 + I, 2, 3)) >>> m Matrix([ [ 0], [1 + I], [ 2], [ 3]]) >>> m.H Matrix([[0, 1  I, 2, 3]])

LDLdecomposition
()[source]¶ Returns the LDL Decomposition (L, D) of matrix A, such that L * D * L.T == A This method eliminates the use of square root. Further this ensures that all the diagonal entries of L are 1. A must be a square, symmetric, positivedefinite and nonsingular matrix.
Examples
>>> A = Matrix(((25, 15, 5), (15, 18, 0), (5, 0, 11))) >>> L, D = A.LDLdecomposition() >>> L Matrix([ [ 1, 0, 0], [ 3/5, 1, 0], [1/5, 1/3, 1]]) >>> D Matrix([ [25, 0, 0], [ 0, 9, 0], [ 0, 0, 9]]) >>> L * D * L.T * A.inv() == eye(A.rows) True
See also

LDLsolve
(rhs)[source]¶ Solves Ax = B using LDL decomposition, for a general square and nonsingular matrix.
For a nonsquare matrix with rows > cols, the least squares solution is returned.
Examples
>>> A = eye(2)*2 >>> B = Matrix([[1, 2], [3, 4]]) >>> A.LDLsolve(B) == B/2 True

LUdecomposition
(iszerofunc=<function _iszero>)[source]¶ Returns the decomposition LU and the row swaps p.
Examples
>>> a = Matrix([[4, 3], [6, 3]]) >>> L, U, _ = a.LUdecomposition() >>> L Matrix([ [ 1, 0], [3/2, 1]]) >>> U Matrix([ [4, 3], [0, 3/2]])

LUdecompositionFF
()[source]¶ Compute a fractionfree LU decomposition.
Returns 4 matrices P, L, D, U such that PA = L D**1 U. If the elements of the matrix belong to some integral domain I, then all elements of L, D and U are guaranteed to belong to I.
 Reference
 W. Zhou & D.J. Jeffrey, “Fractionfree matrix factors: new forms for LU and QR factors”. Frontiers in Computer Science in China, Vol 2, no. 1, pp. 6780, 2008.
See also

LUdecomposition_Simple
(iszerofunc=<function _iszero>)[source]¶ Returns A comprised of L, U (L’s diag entries are 1) and p which is the list of the row swaps (in order).
See also

LUsolve
(rhs, iszerofunc=<function _iszero>)[source]¶ Solve the linear system Ax = rhs for x where A = self.
This is for symbolic matrices, for real or complex ones use mpmath.lu_solve or mpmath.qr_solve.

QRdecomposition
()[source]¶ Return Q, R where A = Q*R, Q is orthogonal and R is upper triangular.
Examples
This is the example from wikipedia:
>>> A = Matrix([[12, 51, 4], [6, 167, 68], [4, 24, 41]]) >>> Q, R = A.QRdecomposition() >>> Q Matrix([ [ 6/7, 69/175, 58/175], [ 3/7, 158/175, 6/175], [2/7, 6/35, 33/35]]) >>> R Matrix([ [14, 21, 14], [ 0, 175, 70], [ 0, 0, 35]]) >>> A == Q*R True
QR factorization of an identity matrix:
>>> A = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> Q, R = A.QRdecomposition() >>> Q Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) >>> R Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]])
See also
cholesky()
,LDLdecomposition()
,LUdecomposition()
,QRsolve()

QRsolve
(b)[source]¶ Solve the linear system ‘Ax = b’.
‘self’ is the matrix ‘A’, the method argument is the vector ‘b’. The method returns the solution vector ‘x’. If ‘b’ is a matrix, the system is solved for each column of ‘b’ and the return value is a matrix of the same shape as ‘b’.
This method is slower (approximately by a factor of 2) but more stable for floatingpoint arithmetic than the LUsolve method. However, LUsolve usually uses an exact arithmetic, so you don’t need to use QRsolve.
This is mainly for educational purposes and symbolic matrices, for real (or complex) matrices use mpmath.qr_solve.

T
¶ Matrix transposition.

adjugate
(method='berkowitz')[source]¶ Returns the adjugate matrix.
Adjugate matrix is the transpose of the cofactor matrix.
https//en.wikipedia.org/wiki/Adjugate
See also

atoms
(*types)[source]¶ Returns the atoms that form the current object.
Examples
>>> Matrix([[x]]) Matrix([[x]]) >>> _.atoms() {x}

berkowitz
()[source]¶ The Berkowitz algorithm.
Given N x N matrix with symbolic content, compute efficiently coefficients of characteristic polynomials of ‘self’ and all its square submatrices composed by removing both ith row and column, without division in the ground domain.
This method is particularly useful for computing determinant, principal minors and characteristic polynomial, when ‘self’ has complicated coefficients e.g. polynomials. Semidirect usage of this algorithm is also important in computing efficiently subresultant PRS.
Assuming that M is a square matrix of dimension N x N and I is N x N identity matrix, then the following following definition of characteristic polynomial is begin used:
charpoly(M) = det(t*I  M)As a consequence, all polynomials generated by Berkowitz algorithm are monic.
>>> M = Matrix([[x, y, z], [1, 0, 0], [y, z, x]])
>>> p, q, r = M.berkowitz()
>>> p # 1 x 1 M's submatrix (1, x)
>>> q # 2 x 2 M's submatrix (1, x, y)
>>> r # 3 x 3 M's submatrix (1, 2*x, x**2  y*z  y, x*y  z**2)
For more information on the implemented algorithm refer to:
 [1] S.J. Berkowitz, On computing the determinant in small
 parallel time using a small number of processors, ACM, Information Processing Letters 18, 1984, pp. 147150
 [2] M. Keber, DivisionFree computation of subresultants
 using Bezout matrices, Tech. Report MPII20061006, Saarbrücken, 2006

berkowitz_charpoly
(x=Dummy('lambda'), simplify=<function simplify>)[source]¶ Computes characteristic polynomial minors using Berkowitz method.
A PurePoly is returned so using different variables for
x
does not affect the comparison or the polynomials:Examples
>>> A = Matrix([[1, 3], [2, 0]]) >>> A.berkowitz_charpoly(x) == A.berkowitz_charpoly(y) True
Specifying
x
is optional; a Dummy with namelambda
is used by default (which looks good when prettyprinted in unicode):>>> A.berkowitz_charpoly().as_expr() _lambda**2  _lambda  6
No test is done to see that
x
doesn’t clash with an existing symbol, so using the default (lambda
) or your own Dummy symbol is the safest option:>>> A = Matrix([[1, 2], [x, 0]]) >>> A.charpoly().as_expr() 2*x + _lambda**2  _lambda >>> A.charpoly(x).as_expr() x**2  3*x
See also

berkowitz_eigenvals
(**flags)[source]¶ Computes eigenvalues of a Matrix using Berkowitz method.
See also

charpoly
(x=Dummy('lambda'), simplify=<function simplify>)¶ Computes characteristic polynomial minors using Berkowitz method.
A PurePoly is returned so using different variables for
x
does not affect the comparison or the polynomials:Examples
>>> A = Matrix([[1, 3], [2, 0]]) >>> A.berkowitz_charpoly(x) == A.berkowitz_charpoly(y) True
Specifying
x
is optional; a Dummy with namelambda
is used by default (which looks good when prettyprinted in unicode):>>> A.berkowitz_charpoly().as_expr() _lambda**2  _lambda  6
No test is done to see that
x
doesn’t clash with an existing symbol, so using the default (lambda
) or your own Dummy symbol is the safest option:>>> A = Matrix([[1, 2], [x, 0]]) >>> A.charpoly().as_expr() 2*x + _lambda**2  _lambda >>> A.charpoly(x).as_expr() x**2  3*x
See also

cholesky
()[source]¶ Returns the Cholesky decomposition L of a matrix A such that L * L.T = A
A must be a square, symmetric, positivedefinite and nonsingular matrix.
Examples
>>> A = Matrix(((25, 15, 5), (15, 18, 0), (5, 0, 11))) >>> A.cholesky() Matrix([ [ 5, 0, 0], [ 3, 3, 0], [1, 1, 3]]) >>> A.cholesky() * A.cholesky().T Matrix([ [25, 15, 5], [15, 18, 0], [5, 0, 11]])
See also

cholesky_solve
(rhs)[source]¶ Solves Ax = B using Cholesky decomposition, for a general square nonsingular matrix. For a nonsquare matrix with rows > cols, the least squares solution is returned.

cofactorMatrix
(method='berkowitz')[source]¶ Return a matrix containing the cofactor of each element.
See also

col_insert
(pos, mti)[source]¶ Insert one or more columns at the given column position.
Examples
>>> M = zeros(3) >>> V = ones(3, 1) >>> M.col_insert(1, V) Matrix([ [0, 1, 0, 0], [0, 1, 0, 0], [0, 1, 0, 0]])
See also

col_join
(bott)[source]¶ Concatenates two matrices along self’s last and bott’s first row
Examples
>>> M = zeros(3) >>> V = ones(1, 3) >>> M.col_join(V) Matrix([ [0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 1, 1]])
See also

condition_number
()[source]¶ Returns the condition number of a matrix.
This is the maximum singular value divided by the minimum singular value
Examples
>>> A = Matrix([[1, 0, 0], [0, 10, 0], [0, 0, S.One/10]]) >>> A.condition_number() 100
See also

cross
(b)[source]¶ Return the cross product of \(self\) and \(b\) relaxing the condition of compatible dimensions: if each has 3 elements, a matrix of the same type and shape as \(self\) will be returned. If \(b\) has the same shape as \(self\) then common identities for the cross product (like \(a x b =  b x a\)) will hold.
See also

det
(method='bareis')[source]¶ Computes the matrix determinant using the method “method”.
 Possible values for “method”:
 bareis … det_bareis berkowitz … berkowitz_det det_LU … det_LU_decomposition
See also

det_LU_decomposition
()[source]¶ Compute matrix determinant using LU decomposition
Note that this method fails if the LU decomposition itself fails. In particular, if the matrix has no inverse this method will fail.
TODO: Implement algorithm for sparse matrices (SFF), Hong R. Lee, B.David Saunders, Fraction Free Gaussian Elimination for Sparse Matrices, In Journal of Symbolic Computation, Volume 19, Issue 5, 1995, Pages 393402, ISSN 07477171, https://www.sciencedirect.com/science/article/pii/S074771718571022X.
See also

det_bareis
()[source]¶ Compute matrix determinant using Bareis’ fractionfree algorithm which is an extension of the well known Gaussian elimination method. This approach is best suited for dense symbolic matrices and will result in a determinant with minimal number of fractions. It means that less term rewriting is needed on resulting formulae.
TODO: Implement algorithm for sparse matrices (SFF), Hong R. Lee, B.David Saunders, Fraction Free Gaussian Elimination for Sparse Matrices, In Journal of Symbolic Computation, Volume 19, Issue 5, 1995, Pages 393402, ISSN 07477171, https://www.sciencedirect.com/science/article/pii/S074771718571022X.
See also

diagonal_solve
(rhs)[source]¶ Solves Ax = B efficiently, where A is a diagonal Matrix, with nonzero diagonal entries.
Examples
>>> A = eye(2)*2 >>> B = Matrix([[1, 2], [3, 4]]) >>> A.diagonal_solve(B) == B/2 True

diagonalize
(reals_only=False, sort=False, normalize=False)[source]¶ Return (P, D), where D is diagonal and
D = P^1 * M * Pwhere M is current matrix.
Examples
>>> m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, 4, 2]) >>> m Matrix([ [1, 2, 0], [0, 3, 0], [2, 4, 2]]) >>> (P, D) = m.diagonalize() >>> D Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> P Matrix([ [1, 0, 1], [ 0, 0, 1], [ 2, 1, 2]]) >>> P.inv() * m * P Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]])
See also

diff
(*args)[source]¶ Calculate the derivative of each element in the matrix.
Examples
>>> M = Matrix([[x, y], [1, 0]]) >>> M.diff(x) Matrix([ [1, 0], [0, 0]])
See also

dot
(b)[source]¶ Return the dot product of Matrix self and b relaxing the condition of compatible dimensions: if either the number of rows or columns are the same as the length of b then the dot product is returned. If self is a row or column vector, a scalar is returned. Otherwise, a list of results is returned (and in that case the number of columns in self must match the length of b).
Examples
>>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> v = [1, 1, 1] >>> M[0, :].dot(v) 6 >>> M[:, 0].dot(v) 12 >>> M.dot(v) [6, 15, 24]
See also

dual
()[source]¶ Returns the dual of a matrix, which is:
\((1/2)*levicivita(i, j, k, l)*M(k, l)\) summed over indices \(k\) and \(l\)
Since the levicivita method is anti_symmetric for any pairwise exchange of indices, the dual of a symmetric matrix is the zero matrix. Strictly speaking the dual defined here assumes that the ‘matrix’ \(M\) is a contravariant anti_symmetric second rank tensor, so that the dual is a covariant second rank tensor.

eigenvals
(**flags)[source]¶ Return eigen values using the berkowitz_eigenvals routine.
Since the roots routine doesn’t always work well with Floats, they will be replaced with Rationals before calling that routine. If this is not desired, set flag
rational
to False.

eigenvects
(**flags)[source]¶ Return list of triples (eigenval, multiplicity, basis).
 The flag
simplify
has two effects:  1) if bool(simplify) is True, as_content_primitive() will be used to tidy up normalization artifacts; 2) if nullspace needs simplification to compute the basis, the simplify flag will be passed on to the nullspace routine which will interpret it there.
If the matrix contains any Floats, they will be changed to Rationals for computation purposes, but the answers will be returned after being evaluated with evalf. If it is desired to removed small imaginary portions during the evalf step, pass a value for the
chop
flag. The flag

expand
(deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints)[source]¶ Apply core.function.expand to each entry of the matrix.
Examples
>>> Matrix(1, 1, [x*(x+1)]) Matrix([[x*(x + 1)]]) >>> _.expand() Matrix([[x**2 + x]])

extract
(rowsList, colsList)[source]¶ Return a submatrix by specifying a list of rows and columns. Negative indices can be given. All indices must be in the range n <= i < n where n is the number of rows or columns.
Examples
>>> m = Matrix(4, 3, range(12)) >>> m Matrix([ [0, 1, 2], [3, 4, 5], [6, 7, 8], [9, 10, 11]]) >>> m.extract([0, 1, 3], [0, 1]) Matrix([ [0, 1], [3, 4], [9, 10]])
Rows or columns can be repeated:
>>> m.extract([0, 0, 1], [1]) Matrix([ [2], [2], [5]])
Every other row can be taken by using range to provide the indices:
>>> m.extract(range(0, m.rows, 2), [1]) Matrix([ [2], [8]])

free_symbols
¶ Returns the free symbols within the matrix.
Examples
>>> Matrix([[x], [1]]).free_symbols {x}

get_diag_blocks
()[source]¶ Obtains the square submatrices on the main diagonal of a square matrix.
Useful for inverting symbolic matrices or solving systems of linear equations which may be decoupled by having a block diagonal structure.
Examples
>>> A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]]) >>> a1, a2, a3 = A.get_diag_blocks() >>> a1 Matrix([ [1, 3], [y, z**2]]) >>> a2 Matrix([[x]]) >>> a3 Matrix([[0]])

has
(*patterns)[source]¶ Test whether any subexpression matches any of the patterns.
Examples
>>> A = Matrix(((1, x), (0.2, 3))) >>> A.has(x) True >>> A.has(y) False >>> A.has(Float) True

classmethod
hstack
(*args)[source]¶ Return a matrix formed by joining args horizontally (i.e. by repeated application of row_join).
Examples
>>> Matrix.hstack(eye(2), 2*eye(2)) Matrix([ [1, 0, 2, 0], [0, 1, 0, 2]])

integrate
(*args)[source]¶ Integrate each element of the matrix.
Examples
>>> M = Matrix([[x, y], [1, 0]]) >>> M.integrate(x) Matrix([ [x**2/2, x*y], [ x, 0]]) >>> M.integrate((x, 0, 2)) Matrix([ [2, 2*y], [2, 0]])

inv
(method=None, **kwargs)[source]¶ Returns the inverse of the matrix.
Parameters: method ({‘GE’, ‘LU’, ‘ADJ’, ‘CH’, ‘LDL’} or None) – Selects algorithm for inversion. For dense matrices available {‘GE’, ‘LU’, ‘ADJ’}, default is ‘GE’. For sparse: {‘CH’, ‘LDL’}, default is ‘LDL’. Raises: ValueError
– If the determinant of the matrix is zero.See also

inv_mod
(m)[source]¶ Returns the inverse of the matrix \(K\) (mod \(m\)), if it exists.
Method to find the matrix inverse of \(K\) (mod \(m\)) implemented in this function:
 Compute \(\mathrm{adj}(K) = \mathrm{cof}(K)^t\), the adjoint matrix of \(K\).
 Compute \(r = 1/\mathrm{det}(K) \pmod m\).
 \(K^{1} = r\cdot \mathrm{adj}(K) \pmod m\).
Examples
>>> A = Matrix(2, 2, [1, 2, 3, 4]) >>> A.inv_mod(5) Matrix([ [3, 1], [4, 2]]) >>> A.inv_mod(3) Matrix([ [1, 1], [0, 1]])

inverse_ADJ
(iszerofunc=<function _iszero>)[source]¶ Calculates the inverse using the adjugate matrix and a determinant.

inverse_GE
(iszerofunc=<function _iszero>)[source]¶ Calculates the inverse using Gaussian elimination.

is_anti_symmetric
(simplify=True)[source]¶ Check if matrix M is an antisymmetric matrix, that is, M is a square matrix with all M[i, j] == M[j, i].
When
simplify=True
(default), the sum M[i, j] + M[j, i] is simplified before testing to see if it is zero. By default, the Diofant simplify function is used. To use a custom function set simplify to a function that accepts a single argument which returns a simplified expression. To skip simplification, set simplify to False but note that although this will be faster, it may induce false negatives.Examples
>>> m = Matrix(2, 2, [0, 1, 1, 0]) >>> m Matrix([ [ 0, 1], [1, 0]]) >>> m.is_anti_symmetric() True >>> x, y = symbols('x y') >>> m = Matrix(2, 3, [0, 0, x, y, 0, 0]) >>> m Matrix([ [ 0, 0, x], [y, 0, 0]]) >>> m.is_anti_symmetric() False
>>> m = Matrix(3, 3, [0, x**2 + 2*x + 1, y, ... (x + 1)**2 , 0, x*y, ... y, x*y, 0])
Simplification of matrix elements is done by default so even though two elements which should be equal and opposite wouldn’t pass an equality test, the matrix is still reported as antisymmetric:
>>> m[0, 1] == m[1, 0] False >>> m.is_anti_symmetric() True
If ‘simplify=False’ is used for the case when a Matrix is already simplified, this will speed things up. Here, we see that without simplification the matrix does not appear antisymmetric:
>>> m.is_anti_symmetric(simplify=False) False
But if the matrix were already expanded, then it would appear antisymmetric and simplification in the is_anti_symmetric routine is not needed:
>>> m = m.expand() >>> m.is_anti_symmetric(simplify=False) True

is_diagonal
()[source]¶ Check if matrix is diagonal, that is matrix in which the entries outside the main diagonal are all zero.
Examples
>>> m = Matrix(2, 2, [1, 0, 0, 2]) >>> m Matrix([ [1, 0], [0, 2]]) >>> m.is_diagonal() True
>>> m = Matrix(2, 2, [1, 1, 0, 2]) >>> m Matrix([ [1, 1], [0, 2]]) >>> m.is_diagonal() False
>>> m = diag(1, 2, 3) >>> m Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> m.is_diagonal() True
See also

is_diagonalizable
(reals_only=False, clear_subproducts=True)[source]¶ Check if matrix is diagonalizable.
If reals_only==True then check that diagonalized matrix consists of the only not complex values.
Some subproducts could be used further in other methods to avoid double calculations, By default (if clear_subproducts==True) they will be deleted.
Examples
>>> m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, 4, 2]) >>> m Matrix([ [1, 2, 0], [0, 3, 0], [2, 4, 2]]) >>> m.is_diagonalizable() True >>> m = Matrix(2, 2, [0, 1, 0, 0]) >>> m Matrix([ [0, 1], [0, 0]]) >>> m.is_diagonalizable() False >>> m = Matrix(2, 2, [0, 1, 1, 0]) >>> m Matrix([ [ 0, 1], [1, 0]]) >>> m.is_diagonalizable() True >>> m.is_diagonalizable(True) False
See also

is_hermitian
¶ Checks if the matrix is Hermitian.
In a Hermitian matrix element i,j is the complex conjugate of element j,i.
Examples
>>> a = Matrix([[1, I], [I, 1]]) >>> a Matrix([ [ 1, I], [I, 1]]) >>> a.is_hermitian True >>> a[0, 0] = 2*I >>> a.is_hermitian False >>> a[0, 0] = x >>> a.is_hermitian >>> a[0, 1] = a[1, 0]*I >>> a.is_hermitian False

is_lower
¶ Check if matrix is a lower triangular matrix. True can be returned even if the matrix is not square.
Examples
>>> m = Matrix(2, 2, [1, 0, 0, 1]) >>> m Matrix([ [1, 0], [0, 1]]) >>> m.is_lower True
>>> m = Matrix(4, 3, [0, 0, 0, 2, 0, 0, 1, 4 , 0, 6, 6, 5]) >>> m Matrix([ [0, 0, 0], [2, 0, 0], [1, 4, 0], [6, 6, 5]]) >>> m.is_lower True
>>> m = Matrix(2, 2, [x**2 + y, y**2 + x, 0, x + y]) >>> m Matrix([ [x**2 + y, x + y**2], [ 0, x + y]]) >>> m.is_lower False
See also

is_lower_hessenberg
¶ Checks if the matrix is in the lowerHessenberg form.
The lower hessenberg matrix has zero entries above the first superdiagonal.
Examples
>>> a = Matrix([[1, 2, 0, 0], [5, 2, 3, 0], [3, 4, 3, 7], [5, 6, 1, 1]]) >>> a Matrix([ [1, 2, 0, 0], [5, 2, 3, 0], [3, 4, 3, 7], [5, 6, 1, 1]]) >>> a.is_lower_hessenberg True
See also

is_nilpotent
()[source]¶ Checks if a matrix is nilpotent.
A matrix B is nilpotent if for some integer k, B**k is a zero matrix.
Examples
>>> a = Matrix([[0, 0, 0], [1, 0, 0], [1, 1, 0]]) >>> a.is_nilpotent() True
>>> a = Matrix([[1, 0, 1], [1, 0, 0], [1, 1, 0]]) >>> a.is_nilpotent() False

is_square
¶ Checks if a matrix is square.
A matrix is square if the number of rows equals the number of columns. The empty matrix is square by definition, since the number of rows and the number of columns are both zero.
Examples
>>> a = Matrix([[1, 2, 3], [4, 5, 6]]) >>> b = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> c = Matrix([]) >>> a.is_square False >>> b.is_square True >>> c.is_square True

is_symbolic
()[source]¶ Checks if any elements contain Symbols.
Examples
>>> M = Matrix([[x, y], [1, 0]]) >>> M.is_symbolic() True

is_symmetric
(simplify=True)[source]¶ Check if matrix is symmetric matrix, that is square matrix and is equal to its transpose.
By default, simplifications occur before testing symmetry. They can be skipped using ‘simplify=False’; while speeding things a bit, this may however induce false negatives.
Examples
>>> m = Matrix(2, 2, [0, 1, 1, 2]) >>> m Matrix([ [0, 1], [1, 2]]) >>> m.is_symmetric() True
>>> m = Matrix(2, 2, [0, 1, 2, 0]) >>> m Matrix([ [0, 1], [2, 0]]) >>> m.is_symmetric() False
>>> m = Matrix(2, 3, [0, 0, 0, 0, 0, 0]) >>> m Matrix([ [0, 0, 0], [0, 0, 0]]) >>> m.is_symmetric() False
>>> m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3]) >>> m Matrix([ [ 1, x**2 + 2*x + 1, y], [(x + 1)**2, 2, 0], [ y, 0, 3]]) >>> m.is_symmetric() True
If the matrix is already simplified, you may speedup is_symmetric() test by using ‘simplify=False’.
>>> m.is_symmetric(simplify=False) False >>> m1 = m.expand() >>> m1.is_symmetric(simplify=False) True

is_upper
¶ Check if matrix is an upper triangular matrix. True can be returned even if the matrix is not square.
Examples
>>> m = Matrix(2, 2, [1, 0, 0, 1]) >>> m Matrix([ [1, 0], [0, 1]]) >>> m.is_upper True
>>> m = Matrix(4, 3, [5, 1, 9, 0, 4 , 6, 0, 0, 5, 0, 0, 0]) >>> m Matrix([ [5, 1, 9], [0, 4, 6], [0, 0, 5], [0, 0, 0]]) >>> m.is_upper True
>>> m = Matrix(2, 3, [4, 2, 5, 6, 1, 1]) >>> m Matrix([ [4, 2, 5], [6, 1, 1]]) >>> m.is_upper False
See also

is_upper_hessenberg
¶ Checks if the matrix is the upperHessenberg form.
The upper hessenberg matrix has zero entries below the first subdiagonal.
Examples
>>> a = Matrix([[1, 4, 2, 3], [3, 4, 1, 7], [0, 2, 3, 4], [0, 0, 1, 3]]) >>> a Matrix([ [1, 4, 2, 3], [3, 4, 1, 7], [0, 2, 3, 4], [0, 0, 1, 3]]) >>> a.is_upper_hessenberg True
See also

is_zero
¶ Checks if a matrix is a zero matrix.
A matrix is zero if every element is zero. A matrix need not be square to be considered zero. The empty matrix is zero by the principle of vacuous truth. For a matrix that may or may not be zero (e.g. contains a symbol), this will be None
Examples
>>> a = Matrix([[0, 0], [0, 0]]) >>> b = zeros(3, 4) >>> c = Matrix([[0, 1], [0, 0]]) >>> d = Matrix([]) >>> e = Matrix([[x, 0], [0, 0]]) >>> a.is_zero True >>> b.is_zero True >>> c.is_zero False >>> d.is_zero True >>> e.is_zero

jacobian
(X)[source]¶ Calculates the Jacobian matrix (derivative of a vectorial function).
Parameters:  self (vector of expressions representing functions f_i(x_1, …, x_n).)
 X (set of x_i’s in order, it can be a list or a Matrix)
 Both self and X can be a row or a column matrix in any order
 (i.e., jacobian() should always work).
Examples
>>> from diofant.abc import rho, phi >>> X = Matrix([rho*cos(phi), rho*sin(phi), rho**2]) >>> Y = Matrix([rho, phi]) >>> X.jacobian(Y) Matrix([ [cos(phi), rho*sin(phi)], [sin(phi), rho*cos(phi)], [ 2*rho, 0]]) >>> X = Matrix([rho*cos(phi), rho*sin(phi)]) >>> X.jacobian(Y) Matrix([ [cos(phi), rho*sin(phi)], [sin(phi), rho*cos(phi)]])

jordan_cells
(calc_transformation=True)[source]¶ Return a list of Jordan cells of current matrix. This list shape Jordan matrix J.
If calc_transformation is specified as False, then transformation P such that
\(J = P^{1} \cdot M \cdot P\)will not be calculated.
Notes
Calculation of transformation P is not implemented yet.
Examples
>>> m = Matrix(4, 4, [ ... 6, 5, 2, 3, ... 3, 1, 3, 3, ... 2, 1, 2, 3, ... 1, 1, 5, 5])
>>> P, Jcells = m.jordan_cells() >>> Jcells[0] Matrix([ [2, 1], [0, 2]]) >>> Jcells[1] Matrix([ [2, 1], [0, 2]])
See also

jordan_form
(calc_transformation=True)[source]¶ Return Jordan form J of current matrix.
Also the transformation P such that
\(J = P^{1} \cdot M \cdot P\)and the jordan blocks forming J will be calculated.
Examples
>>> m = Matrix([ ... [ 6, 5, 2, 3], ... [3, 1, 3, 3], ... [ 2, 1, 2, 3], ... [1, 1, 5, 5]]) >>> P, J = m.jordan_form() >>> J Matrix([ [2, 1, 0, 0], [0, 2, 0, 0], [0, 0, 2, 1], [0, 0, 0, 2]])
See also

key2bounds
(keys)[source]¶ Converts a key with potentially mixed types of keys (integer and slice) into a tuple of ranges and raises an error if any index is out of self’s range.
See also

key2ij
(key)[source]¶ Converts key into canonical form, converting integers or indexable items into valid integers for self’s range or returning slices unchanged.
See also

limit
(*args)[source]¶ Calculate the limit of each element in the matrix.
Examples
>>> M = Matrix([[x, y], [1, 0]]) >>> M.limit(x, 2) Matrix([ [2, y], [1, 0]])
See also

multiply_elementwise
(b)[source]¶ Return the Hadamard product (elementwise product) of A and B
Examples
>>> A = Matrix([[0, 1, 2], [3, 4, 5]]) >>> B = Matrix([[1, 10, 100], [100, 10, 1]]) >>> A.multiply_elementwise(B) Matrix([ [ 0, 10, 200], [300, 40, 5]])
See also

n
(dps=15, **options)¶ Apply evalf() to each element of self.

norm
(ord=None)[source]¶ Return the Norm of a Matrix or Vector. In the simplest case this is the geometric size of the vector Other norms can be specified by the ord parameter
ord norm for matrices norm for vectors None Frobenius norm 2norm ‘fro’ Frobenius norm  does not exist
inf – max(abs(x)) inf – min(abs(x)) 1 – as below 1 – as below 2 2norm (largest sing. value) as below 2 smallest singular value as below other  does not exist
sum(abs(x)**ord)**(1./ord) Examples
>>> x = Symbol('x', real=True) >>> v = Matrix([cos(x), sin(x)]) >>> trigsimp(v.norm()) 1 >>> v.norm(10) (sin(x)**10 + cos(x)**10)**(1/10) >>> A = Matrix([[1, 1], [1, 1]]) >>> A.norm(2) # Spectral norm (max of Ax/x under 2vectornorm) 2 >>> A.norm(2) # Inverse spectral norm (smallest singular value) 0 >>> A.norm() # Frobenius Norm 2 >>> Matrix([1, 2]).norm(oo) 2 >>> Matrix([1, 2]).norm(oo) 1
See also

nullspace
(simplify=False)[source]¶ Returns list of vectors (Matrix objects) that span nullspace of self

permuteBkwd
(perm)[source]¶ Permute the rows of the matrix with the given permutation in reverse.
Examples
>>> M = eye(3) >>> M.permuteBkwd([[0, 1], [0, 2]]) Matrix([ [0, 1, 0], [0, 0, 1], [1, 0, 0]])
See also

permuteFwd
(perm)[source]¶ Permute the rows of the matrix with the given permutation.
Examples
>>> M = eye(3) >>> M.permuteFwd([[0, 1], [0, 2]]) Matrix([ [0, 0, 1], [1, 0, 0], [0, 1, 0]])
See also

pinv
()[source]¶ Calculate the MoorePenrose pseudoinverse of the matrix.
The MoorePenrose pseudoinverse exists and is unique for any matrix. If the matrix is invertible, the pseudoinverse is the same as the inverse.
Examples
>>> Matrix([[1, 2, 3], [4, 5, 6]]).pinv() Matrix([ [17/18, 4/9], [ 1/9, 1/9], [ 13/18, 2/9]])
References

pinv_solve
(B, arbitrary_matrix=None)[source]¶ Solve Ax = B using the MoorePenrose pseudoinverse.
There may be zero, one, or infinite solutions. If one solution exists, it will be returned. If infinite solutions exist, one will be returned based on the value of arbitrary_matrix. If no solutions exist, the leastsquares solution is returned.
Parameters:  B (Matrix) – The right hand side of the equation to be solved for. Must have the same number of rows as matrix A.
 arbitrary_matrix (Matrix) – If the system is underdetermined (e.g. A has more columns than
rows), infinite solutions are possible, in terms of an arbitrary
matrix. This parameter may be set to a specific matrix to use
for that purpose; if so, it must be the same shape as x, with as
many rows as matrix A has columns, and as many columns as matrix
B. If left as None, an appropriate matrix containing dummy
symbols in the form of
wn_m
will be used, with n and m being row and column position of each symbol.
Returns: x (Matrix) – The matrix that will satisfy Ax = B. Will have as many rows as matrix A has columns, and as many columns as matrix B.
Examples
>>> A = Matrix([[1, 2, 3], [4, 5, 6]]) >>> B = Matrix([7, 8]) >>> A.pinv_solve(B) Matrix([ [ _w0_0/6  _w1_0/3 + _w2_0/6  55/18], [_w0_0/3 + 2*_w1_0/3  _w2_0/3 + 1/9], [ _w0_0/6  _w1_0/3 + _w2_0/6 + 59/18]]) >>> A.pinv_solve(B, arbitrary_matrix=Matrix([0, 0, 0])) Matrix([ [55/18], [ 1/9], [ 59/18]])
See also
lower_triangular_solve()
,upper_triangular_solve()
,cholesky_solve()
,diagonal_solve()
,LDLsolve()
,LUsolve()
,QRsolve()
,pinv()
Notes
This may return either exact solutions or least squares solutions. To determine which, check
A * A.pinv() * B == B
. It will be True if exact solutions exist, and False if only a leastsquares solution exists. Be aware that the left hand side of that equation may need to be simplified to correctly compare to the right hand side.References

print_nonzero
(symb='X')[source]¶ Shows location of nonzero entries for fast shape lookup.
Examples
>>> m = Matrix(2, 3, lambda i, j: i*3+j) >>> m Matrix([ [0, 1, 2], [3, 4, 5]]) >>> m.print_nonzero() [ XX] [XXX] >>> m = eye(4) >>> m.print_nonzero("x") [x ] [ x ] [ x ] [ x]

project
(v)[source]¶ Return the projection of
self
onto the line containingv
.Examples
>>> V = Matrix([sqrt(3)/2, S.Half]) >>> x = Matrix([[1, 0]]) >>> V.project(x) Matrix([[sqrt(3)/2, 0]]) >>> V.project(x) Matrix([[sqrt(3)/2, 0]])

rank
(iszerofunc=<function _iszero>, simplify=False)[source]¶ Returns the rank of a matrix
>>> m = Matrix([[1, 2], [x, 1  1/x]]) >>> m.rank() 2 >>> n = Matrix(3, 3, range(1, 10)) >>> n.rank() 2

replace
(F, G)[source]¶ Replaces Function F in Matrix entries with Function G.
Examples
>>> F, G = symbols('F, G', cls=Function) >>> M = Matrix(2, 2, lambda i, j: F(i+j)) ; M Matrix([ [F(0), F(1)], [F(1), F(2)]]) >>> N = M.replace(F, G) >>> N Matrix([ [G(0), G(1)], [G(1), G(2)]])

row_insert
(pos, mti)[source]¶ Insert one or more rows at the given row position.
Examples
>>> M = zeros(3) >>> V = ones(1, 3) >>> M.row_insert(1, V) Matrix([ [0, 0, 0], [1, 1, 1], [0, 0, 0], [0, 0, 0]])
See also

row_join
(rhs)[source]¶ Concatenates two matrices along self’s last and rhs’s first column
Examples
>>> M = zeros(3) >>> V = ones(3, 1) >>> M.row_join(V) Matrix([ [0, 0, 0, 1], [0, 0, 0, 1], [0, 0, 0, 1]])
See also

rref
(iszerofunc=<function _iszero>, simplify=False)[source]¶ Return reduced rowechelon form of matrix and indices of pivot vars.
To simplify elements before finding nonzero pivots set simplify=True (to use the default Diofant simplify function) or pass a custom simplify function.
Examples
>>> m = Matrix([[1, 2], [x, 1  1/x]]) >>> m.rref() (Matrix([ [1, 0], [0, 1]]), [0, 1])

shape
¶ The shape (dimensions) of the matrix as the 2tuple (rows, cols).
Examples
>>> M = zeros(2, 3) >>> M.shape (2, 3) >>> M.rows 2 >>> M.cols 3

simplify
(ratio=1.7, measure=<function count_ops>)[source]¶ Apply simplify to each element of the matrix.
Examples
>>> SparseMatrix(1, 1, [x*sin(y)**2 + x*cos(y)**2]) Matrix([[x*sin(y)**2 + x*cos(y)**2]]) >>> _.simplify() Matrix([[x]])

singular_values
()[source]¶ Compute the singular values of a Matrix
Examples
>>> x = Symbol('x', real=True) >>> A = Matrix([[0, 1, 0], [0, x, 0], [1, 0, 0]]) >>> A.singular_values() [sqrt(x**2 + 1), 1, 0]
See also

solve_least_squares
(rhs, method='CH')[source]¶ Return the leastsquare fit to the data.
By default the cholesky_solve routine is used (method=’CH’); other methods of matrix inversion can be used.
Examples
>>> A = Matrix([1, 2, 3]) >>> B = Matrix([2, 3, 4]) >>> S = Matrix(A.row_join(B)) >>> S Matrix([ [1, 2], [2, 3], [3, 4]])
If each line of S represent coefficients of Ax + By and x and y are [2, 3] then S*xy is:
>>> r = S*Matrix([2, 3]); r Matrix([ [ 8], [13], [18]])
But let’s add 1 to the middle value and then solve for the leastsquares value of xy:
>>> xy = S.solve_least_squares(Matrix([8, 14, 18])); xy Matrix([ [ 5/3], [10/3]])
The error is given by S*xy  r:
>>> S*xy  r Matrix([ [1/3], [1/3], [1/3]]) >>> _.norm().evalf(2) 0.58
If a different xy is used, the norm will be higher:
>>> xy += ones(2, 1)/10 >>> (S*xy  r).norm().evalf(2) 1.5
See also

subs
(*args, **kwargs)[source]¶ Return a new matrix with subs applied to each entry.
Examples
>>> SparseMatrix(1, 1, [x]) Matrix([[x]]) >>> _.subs({x: y}) Matrix([[y]]) >>> Matrix(_).subs({y: x}) Matrix([[x]])

table
(printer, rowstart='[', rowend=']', rowsep='\n', colsep=', ', align='right')[source]¶ String form of Matrix as a table.
printer
is the printer to use for on the elements (generally something like StrPrinter())rowstart
is the string used to start each row (by default ‘[‘).rowend
is the string used to end each row (by default ‘]’).rowsep
is the string used to separate rows (by default a newline).colsep
is the string used to separate columns (by default ‘, ‘).align
defines how the elements are aligned. Must be one of ‘left’, ‘right’, or ‘center’. You can also use ‘<’, ‘>’, and ‘^’ to mean the same thing, respectively.This is used by the string printer for Matrix.
Examples
>>> from diofant.printing.str import StrPrinter >>> M = Matrix([[1, 2], [33, 4]]) >>> printer = StrPrinter() >>> M.table(printer) '[ 1, 2]\n[33, 4]' >>> print(M.table(printer)) [ 1, 2] [33, 4] >>> print(M.table(printer, rowsep=',\n')) [ 1, 2], [33, 4] >>> print('[%s]' % M.table(printer, rowsep=',\n')) [[ 1, 2], [33, 4]] >>> print(M.table(printer, colsep=' ')) [ 1 2] [33 4] >>> print(M.table(printer, align='center')) [ 1 , 2] [33, 4] >>> print(M.table(printer, rowstart='{', rowend='}')) { 1, 2} {33, 4}

vec
()[source]¶ Return the Matrix converted into a one column matrix by stacking columns
Examples
>>> m = Matrix([[1, 3], [2, 4]]) >>> m Matrix([ [1, 3], [2, 4]]) >>> m.vec() Matrix([ [1], [2], [3], [4]])
See also

vech
(diagonal=True, check_symmetry=True)[source]¶ Return the unique elements of a symmetric Matrix as a one column matrix by stacking the elements in the lower triangle.
Arguments: diagonal – include the diagonal cells of self or not check_symmetry – checks symmetry of self but not completely reliably
Examples
>>> m = Matrix([[1, 2], [2, 3]]) >>> m Matrix([ [1, 2], [2, 3]]) >>> m.vech() Matrix([ [1], [2], [3]]) >>> m.vech(diagonal=False) Matrix([[2]])
See also

Matrix Exceptions Reference¶
Matrix Functions Reference¶

diofant.matrices.matrices.
classof
(A, B)[source]¶ Get the type of the result when combining matrices of different types.
Currently the strategy is that immutability is contagious.
Examples
>>> M = Matrix([[1, 2], [3, 4]]) # a Mutable Matrix >>> IM = ImmutableMatrix([[1, 2], [3, 4]]) >>> classof(M, IM) <class 'diofant.matrices.immutable.ImmutableMatrix'>

diofant.matrices.dense.
matrix_multiply_elementwise
(A, B)[source]¶ Return the Hadamard product (elementwise product) of A and B
>>> A = Matrix([[0, 1, 2], [3, 4, 5]]) >>> B = Matrix([[1, 10, 100], [100, 10, 1]]) >>> matrix_multiply_elementwise(A, B) Matrix([ [ 0, 10, 200], [300, 40, 5]])

diofant.matrices.dense.
zeros
(r, c=None, cls=None)[source]¶ Returns a matrix of zeros with
r
rows andc
columns; ifc
is omitted a square matrix will be returned.

diofant.matrices.dense.
ones
(r, c=None)[source]¶ Returns a matrix of ones with
r
rows andc
columns; ifc
is omitted a square matrix will be returned.

diofant.matrices.dense.
diag
(*values, **kwargs)[source]¶ Create a sparse, diagonal matrix from a list of diagonal values.
Notes
When arguments are matrices they are fitted in resultant matrix.
The returned matrix is a mutable, dense matrix. To make it a different type, send the desired class for keyword
cls
.Examples
>>> diag(1, 2, 3) Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> diag(*[1, 2, 3]) Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]])
The diagonal elements can be matrices; diagonal filling will continue on the diagonal from the last element of the matrix:
>>> a = Matrix([x, y, z]) >>> b = Matrix([[1, 2], [3, 4]]) >>> c = Matrix([[5, 6]]) >>> diag(a, 7, b, c) Matrix([ [x, 0, 0, 0, 0, 0], [y, 0, 0, 0, 0, 0], [z, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0], [0, 0, 1, 2, 0, 0], [0, 0, 3, 4, 0, 0], [0, 0, 0, 0, 5, 6]])
When diagonal elements are lists, they will be treated as arguments to Matrix:
>>> diag([1, 2, 3], 4) Matrix([ [1, 0], [2, 0], [3, 0], [0, 4]]) >>> diag([[1, 2, 3]], 4) Matrix([ [1, 2, 3, 0], [0, 0, 0, 4]])
A given band off the diagonal can be made by padding with a vertical or horizontal “kerning” vector:
>>> hpad = ones(0, 2) >>> vpad = ones(2, 0) >>> diag(vpad, 1, 2, 3, hpad) + diag(hpad, 4, 5, 6, vpad) Matrix([ [0, 0, 4, 0, 0], [0, 0, 0, 5, 0], [1, 0, 0, 0, 6], [0, 2, 0, 0, 0], [0, 0, 3, 0, 0]])
The type is mutable by default but can be made immutable by setting the
mutable
flag to False:>>> type(diag(1)) <class 'diofant.matrices.dense.MutableDenseMatrix'> >>> type(diag(1, cls=ImmutableMatrix)) <class 'diofant.matrices.immutable.ImmutableMatrix'>
See also

diofant.matrices.dense.
jordan_cell
(eigenval, n)[source]¶ Create matrix of Jordan cell kind:
Examples
>>> jordan_cell(x, 4) Matrix([ [x, 1, 0, 0], [0, x, 1, 0], [0, 0, x, 1], [0, 0, 0, x]])

diofant.matrices.dense.
hessian
(f, varlist, constraints=[])[source]¶ Compute Hessian matrix for a function f wrt parameters in varlist which may be given as a sequence or a row/column vector. A list of constraints may optionally be given.
Examples
>>> f = Function('f')(x, y) >>> g1 = Function('g')(x, y) >>> g2 = x**2 + 3*y >>> pprint(hessian(f, (x, y), [g1, g2]), use_unicode=False) [ d d ] [ 0 0 (g(x, y)) (g(x, y)) ] [ dx dy ] [ ] [ 0 0 2*x 3 ] [ ] [ 2 2 ] [d d d ] [(g(x, y)) 2*x (f(x, y)) (f(x, y))] [dx 2 dy dx ] [ dx ] [ ] [ 2 2 ] [d d d ] [(g(x, y)) 3 (f(x, y)) (f(x, y)) ] [dy dy dx 2 ] [ dy ]
References
https//en.wikipedia.org/wiki/Hessian_matrix

diofant.matrices.dense.
GramSchmidt
(vlist, orthonormal=False)[source]¶ Apply the GramSchmidt process to a set of vectors.
see: https//en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process

diofant.matrices.dense.
wronskian
(functions, var, method='bareis')[source]¶ Compute Wronskian for [] of functions
 f1 f2 ... fn   f1' f2' ... fn'   . . . .  W(f1, ..., fn) =  . . . .   . . . .   (n) (n) (n)   D (f1) D (f2) ... D (fn) 
see: https//en.wikipedia.org/wiki/Wronskian

diofant.matrices.dense.
casoratian
(seqs, n, zero=True)[source]¶ Given linear difference operator L of order ‘k’ and homogeneous equation Ly = 0 we want to compute kernel of L, which is a set of ‘k’ sequences: a(n), b(n), … z(n).
Solutions of L are linearly independent iff their Casoratian, denoted as C(a, b, …, z), do not vanish for n = 0.
Casoratian is defined by k x k determinant:
+ a(n) b(n) . . . z(n) +  a(n+1) b(n+1) . . . z(n+1)   . . . .   . . . .   . . . .  + a(n+k1) b(n+k1) . . . z(n+k1) +
It proves very useful in rsolve_hyper() where it is applied to a generating set of a recurrence to factor out linearly dependent solutions and return a basis:
>>> n = Symbol('n', integer=True)
Exponential and factorial are linearly independent:
>>> casoratian([2**n, factorial(n)], n) != 0 True

diofant.matrices.dense.
randMatrix
(r, c=None, min=0, max=99, seed=None, symmetric=False, percent=100)[source]¶ Create random matrix with dimensions
r
xc
. Ifc
is omitted the matrix will be square. Ifsymmetric
is True the matrix must be square. Ifpercent
is less than 100 then only approximately the given percentage of elements will be nonzero.Examples
>>> randMatrix(3) # doctest:+SKIP [25, 45, 27] [44, 54, 9] [23, 96, 46] >>> randMatrix(3, 2) # doctest:+SKIP [87, 29] [23, 37] [90, 26] >>> randMatrix(3, 3, 0, 2) # doctest:+SKIP [0, 2, 0] [2, 0, 1] [0, 0, 1] >>> randMatrix(3, symmetric=True) # doctest:+SKIP [85, 26, 29] [26, 71, 43] [29, 43, 57] >>> A = randMatrix(3, seed=1) >>> B = randMatrix(3, seed=2) >>> A == B # doctest:+SKIP False >>> A == randMatrix(3, seed=1) True >>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP [0, 68, 43] [0, 68, 0] [0, 91, 34]
Numpy Utility Functions Reference¶

diofant.matrices.dense.
list2numpy
(l, dtype=<class 'object'>)[source]¶ Converts python list of Diofant expressions to a NumPy array.

diofant.matrices.dense.
matrix2numpy
(m, dtype=<class 'object'>)[source]¶ Converts Diofant’s matrix to a NumPy array.
See also

diofant.matrices.dense.
symarray
(prefix, shape, **kwargs)[source]¶ Create a numpy ndarray of symbols (as an object array).
The created symbols are named
prefix_i1_i2_
… You should thus provide a nonempty prefix if you want your symbols to be unique for different output arrays, as Diofant symbols with identical names are the same object.Parameters:  prefix (string) – A prefix prepended to the name of every symbol.
 shape (int or tuple) – Shape of the created array. If an int, the array is onedimensional; for more than one dimension the shape must be a tuple.
 **kwargs (dict) – keyword arguments passed on to Symbol
Examples
These doctests require numpy.
>>> symarray('', 3) [_0 _1 _2]
If you want multiple symarrays to contain distinct symbols, you must provide unique prefixes:
>>> a = symarray('', 3) >>> b = symarray('', 3) >>> a[0] == b[0] True >>> a = symarray('a', 3) >>> b = symarray('b', 3) >>> a[0] == b[0] False
Creating symarrays with a prefix:
>>> symarray('a', 3) [a_0 a_1 a_2]
For more than one dimension, the shape must be given as a tuple:
>>> symarray('a', (2, 3)) [[a_0_0 a_0_1 a_0_2] [a_1_0 a_1_1 a_1_2]] >>> symarray('a', (2, 3, 2)) [[[a_0_0_0 a_0_0_1] [a_0_1_0 a_0_1_1] [a_0_2_0 a_0_2_1]] <BLANKLINE> [[a_1_0_0 a_1_0_1] [a_1_1_0 a_1_1_1] [a_1_2_0 a_1_2_1]]]
For setting assumptions of the underlying Symbols:
>>> [s.is_real for s in symarray('a', 2, real=True)] [True, True]

diofant.matrices.dense.
rot_axis1
(theta)[source]¶ Returns a rotation matrix for a rotation of theta (in radians) about the 1axis.
Examples
A rotation of pi/3 (60 degrees):
>>> theta = pi/3 >>> rot_axis1(theta) Matrix([ [1, 0, 0], [0, 1/2, sqrt(3)/2], [0, sqrt(3)/2, 1/2]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis1(pi/2) Matrix([ [1, 0, 0], [0, 0, 1], [0, 1, 0]])
See also
diofant.matrices.dense.rot_axis2()
 Returns a rotation matrix for a rotation of theta (in radians) about the 2axis
diofant.matrices.dense.rot_axis3()
 Returns a rotation matrix for a rotation of theta (in radians) about the 3axis

diofant.matrices.dense.
rot_axis2
(theta)[source]¶ Returns a rotation matrix for a rotation of theta (in radians) about the 2axis.
Examples
A rotation of pi/3 (60 degrees):
>>> theta = pi/3 >>> rot_axis2(theta) Matrix([ [ 1/2, 0, sqrt(3)/2], [ 0, 1, 0], [sqrt(3)/2, 0, 1/2]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis2(pi/2) Matrix([ [0, 0, 1], [0, 1, 0], [1, 0, 0]])
See also
diofant.matrices.dense.rot_axis1()
 Returns a rotation matrix for a rotation of theta (in radians) about the 1axis
diofant.matrices.dense.rot_axis3()
 Returns a rotation matrix for a rotation of theta (in radians) about the 3axis

diofant.matrices.dense.
rot_axis3
(theta)[source]¶ Returns a rotation matrix for a rotation of theta (in radians) about the 3axis.
Examples
A rotation of pi/3 (60 degrees):
>>> theta = pi/3 >>> rot_axis3(theta) Matrix([ [ 1/2, sqrt(3)/2, 0], [sqrt(3)/2, 1/2, 0], [ 0, 0, 1]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis3(pi/2) Matrix([ [ 0, 1, 0], [1, 0, 0], [ 0, 0, 1]])
See also
diofant.matrices.dense.rot_axis1()
 Returns a rotation matrix for a rotation of theta (in radians) about the 1axis
diofant.matrices.dense.rot_axis2()
 Returns a rotation matrix for a rotation of theta (in radians) about the 2axis