WSMatrix E¶
jwsagg.spad line 341 [edit on github]
E: WSRing
Julia Wolfram Symbolic matrices using the MathLink Julia package.
- #: % -> NonNegativeInteger
from Aggregate
- #: % -> WSInteger
from WSAggregate WSVector E
- *: (%, %) -> %
a * b
is the matrix multiplication, not the element-wise multiplication.WS
error if dimensions are incompatible.
- *: (%, E) -> %
x * r
is the right scalar multiple of the scalarr
and the matrixx
.
- *: (%, WSVector E) -> %
x * c
is the product of the matrixx
and the column vectorc
. Error: if the dimensions are incompatible.- *: (%, WSVector E) -> WSVector E
from MatrixCategory(E, WSVector E, WSVector E)
- *: (E, %) -> %
r*x
is the left scalar multiple of the scalarr
and the matrixx
.
- *: (Integer, %) -> %
n*a
scale the matrix a byn
.
- *: (WSVector E, %) -> %
r * x
is the product of the row vectorr
and the matrixx
. Error: if the dimensions are incompatible.- *: (WSVector E, %) -> WSVector E
from MatrixCategory(E, WSVector E, WSVector E)
- +: (%, %) -> %
a + b
is the matrix addition.WS
error if dimensions are incompatible.
- -: % -> %
-a
negates each elements of the matrix a.
- -: (%, %) -> %
a - b
is the matrix substraction.WS
error if dimensions are incompatible.
- /: (%, E) -> % if E has Field
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- ^: (%, NonNegativeInteger) -> %
a^n
is the matrix power by the non negative integern
.
- ^: (%, PositiveInteger) -> %
a^n
is the matrix power by the positive integern
.
- accumulate: % -> % if WSVector E has WSNumber
from WSAggregate WSVector E
- adjoint: % -> %
adjoint(m)
returns the adjoint ofm
, i.e. the conjugate transposition ofm
.
- adjugate: % -> %
adjugate(m)
returns the adjugate of squarem
.
- antisymmetric?: % -> Boolean
antisymmetric?(m)
returnstrue
if the matrixm
is square and antisymmetric (i.e.m[i, j] = -m[j, i]
for alli
andj
) andfalse
otherwise.
- any?: (E -> Boolean, %) -> Boolean
from HomogeneousAggregate E
- append: (%, WSVector E) -> %
from WSAggregate WSVector E
- blockConcat: List List % -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- blockSplit: (%, List NonNegativeInteger, List NonNegativeInteger) -> List List %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- blockSplit: (%, PositiveInteger, PositiveInteger) -> List List %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- cholesky: % -> Record(l: %, u: %)
cholesky(m)
computes the Cholesky decomposition of the square matrixm
.
- coerce: % -> JLObject
from JLObjectType
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: % -> WSExpression
from WSAggregate WSVector E
- coerce: List WSVector E -> %
from WSAggregate WSVector E
- coerce: WSVector E -> %
from MatrixCategory(E, WSVector E, WSVector E)
- column: (%, Integer) -> %
column(mat,i)
returns thei
-th column.- column: (%, Integer) -> WSVector E
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- columnSpace: % -> List WSVector E if E has EuclideanDomain
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- convert: % -> String
from ConvertibleTo String
- count: (E -> Boolean, %) -> NonNegativeInteger
from HomogeneousAggregate E
- count: (E, %) -> NonNegativeInteger
from HomogeneousAggregate E
- delete: (%, WSList WSInteger) -> %
from WSAggregate WSVector E
- determinant: % -> E if E has CommutativeRing
from MatrixCategory(E, WSVector E, WSVector E)
- diagonal?: % -> Boolean
diagonal?(m)
returnstrue
if the matrixm
is square and diagonal (i.e. all entries ofm
not on the diagonal are zero) andfalse
otherwise.
- diagonal: % -> WSVector E
diagonal(m)
returns the diagonal elements ofm
as a vector.
- diagonalMatrix: List % -> %
from MatrixCategory(E, WSVector E, WSVector E)
- diagonalMatrix: List E -> %
from MatrixCategory(E, WSVector E, WSVector E)
- diagonalMatrix: WSList E -> %
diagonalMatrix(l)
returns a diagonal matrix with elements ofl
as diagonal elements.
- differences: % -> % if WSVector E has WSNumber
from WSAggregate WSVector E
- dimensions: % -> WSList WSInteger
from WSAggregate WSVector E
- eigenSystem: % -> Record(values: WSVector WSExpression, vectors: WSMatrix WSExpression)
eigenSystem(m)
computes the spectral decomposition of the square matrixm
. It returns in a Record, the eigenvalues, selector ‘values’, and the eigenvectors selector ‘vectors’.
- eigenvalues: % -> WSVector WSExpression
eigenvalues(m)
returns the eigenvalues of the square matrixm
.
- eigenvectors: % -> WSMatrix WSExpression
eigenvectors(m)
returns the eigenvectors of the square matrixm
.
- elt: (%, Integer) -> WSVector E
from WSAggregate WSVector E
- elt: (%, Integer, Integer) -> E
elt(mat, m, n)
returns the element (m
,n
) of the matrixmat
.- elt: (%, Integer, Integer, E) -> E
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, Integer, List Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, Integer, List Segment Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, List Integer, Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, List Integer, List Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, List Integer, Segment Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, List Segment Integer, Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, List Segment Integer, List Segment Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, List Segment Integer, Segment Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, Segment Integer, List Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, Segment Integer, List Segment Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- elt: (%, Segment Integer, Segment Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- eval: (%, E, E) -> % if E has Evalable E
from InnerEvalable(E, E)
- eval: (%, Equation E) -> % if E has Evalable E
from Evalable E
- eval: (%, List E, List E) -> % if E has Evalable E
from InnerEvalable(E, E)
- eval: (%, List Equation E) -> % if E has Evalable E
from Evalable E
- every?: (E -> Boolean, %) -> Boolean
from HomogeneousAggregate E
- exquo: (%, E) -> Union(%, failed) if E has IntegralDomain
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- extract: (%, WSExpression) -> %
from WSAggregate WSVector E
- fill!: (%, E) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- first: % -> WSVector E
from WSAggregate WSVector E
- generalizedInverse: % -> %
generalizedInverse(m)
computes the pseudo inverse ofm
also knows as Moore-Penrose inverse.
- hash: % -> SingleInteger if E has Hashable
from Hashable
- hashUpdate!: (HashState, %) -> HashState if E has Hashable
from Hashable
- hermitian?: % -> Boolean
hermitian?(m)
checks whether or notm
is hermitian.
- hessenberg: % -> Record(p: %, h: %)
hessenberg(m)
computes the Hessenberg decomposition of the square matrixm
.
- horizConcat: (%, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- horizConcat: List % -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- horizSplit: (%, List NonNegativeInteger) -> List %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- horizSplit: (%, PositiveInteger) -> List %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- insert: (%, WSVector E, WSInteger) -> %
from WSAggregate WSVector E
- intersection: (%, %) -> %
from WSAggregate WSVector E
- inverse: % -> %
inverse(m)
computes the inverse ofm
. For example: example{m:=jWSMatrix(”{{a, b}, {b, a}}”)@WSMAT(WSEXPR)} example{inverse m}- inverse: % -> Union(%, failed) if E has Field
from MatrixCategory(E, WSVector E, WSVector E)
- invertIfCan: % -> Union(%, failed) if E has IntegralDomain
invertIfCan(m)
returns the inverse of the matrixm
. If the matrix is not invertible, “failed” is returned. Error: if the matrix is not square.
- jlAbout: % -> Void
from JLObjectType
- jlApply: (String, %) -> %
from JLObjectType
- jlApply: (String, %, %) -> %
from JLObjectType
- jlApply: (String, %, %, %) -> %
from JLObjectType
- jlApply: (String, %, %, %, %) -> %
from JLObjectType
- jlApply: (String, %, %, %, %, %) -> %
from JLObjectType
- jlDisplay: % -> Void
from JLObjectType
- jlDump: JLObject -> Void
from JLObjectType
- jlId: % -> JLInt64
from JLObjectType
- jlNumeric: % -> %
from WSObject
- jlNumeric: (%, PositiveInteger) -> %
from WSObject
- jlObject: () -> String
from JLObjectType
- jlRef: % -> SExpression
from JLObjectType
- jlref: String -> %
from JLObjectType
- jlSymbolic: % -> String
from WSObject
- jlType: % -> String
from JLObjectType
- join: (%, %) -> %
from WSAggregate WSVector E
- jordan: % -> Record(s: %, j: %)
jordan(m)
computes the Jordan decomposition of of the square matrixm
.
- jWSAggregate: List WSVector E -> %
from WSAggregate WSVector E
- jWSInterpret: (String, String) -> %
from WSObject
- jWSMatrix: String -> %
jWSMatrix(str)
constructsstr
as a WSMatrix.str
must be in theWS
language (WS
list ofWS
list(s
)).
- kronecker_prod1: (%, Integer, List List NonNegativeInteger, List %, NonNegativeInteger, NonNegativeInteger, Union(E, one)) -> Void
from MatrixCategory(E, WSVector E, WSVector E)
- kroneckerProduct: (%, %) -> %
from MatrixCategory(E, WSVector E, WSVector E)
- kroneckerProduct: List % -> %
from MatrixCategory(E, WSVector E, WSVector E)
- kroneckerSum: (%, %) -> %
from MatrixCategory(E, WSVector E, WSVector E)
- kroneckerSum: List % -> %
from MatrixCategory(E, WSVector E, WSVector E)
- last: % -> WSVector E
from WSAggregate WSVector E
- latex: % -> String
from SetCategory
- length: % -> WSInteger
from WSAggregate WSVector E
- less?: (%, NonNegativeInteger) -> Boolean
from Aggregate
- linearSolve: (%, %) -> %
linearSolve(mat, b)
findsx
, solution of the equationmat
*x
=b
.
- listOfLists: % -> List List E
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- lu: % -> Record(lu: %, p: %, c: %)
lu(m)
computes the lU decomposition of the matrixm
.
- map!: (E -> E, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- map: ((E, E) -> E, %, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- map: ((E, E) -> E, %, %, E) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- map: (E -> E, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- matrix: (NonNegativeInteger, NonNegativeInteger, (Integer, Integer) -> E) -> %
from MatrixCategory(E, WSVector E, WSVector E)
- matrix: List List E -> %
from MatrixCategory(E, WSVector E, WSVector E)
- matrix: WSVector E -> %
matrix(vec)
returnsvec
as a matrix (inplace coercion).
- matrixFunction: (WSExpression, %) -> WSMatrix WSExpression
matrixFunction(func, m)
applies the matrix functionfunc
, if available, to the matrixm
. Note that it is not an element-wise operation. For example: example{m := jWSMatrix(”{{1.2, 1.7},{1.12, -2.1}}”)@WSMAT(WSREAL)} example{matrixFunction(“Sqrt”, m)}
- max: % -> E if E has OrderedSet
from HomogeneousAggregate E
- max: ((E, E) -> Boolean, %) -> E
from HomogeneousAggregate E
- maxColIndex: % -> Integer
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- maxRowIndex: % -> Integer
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- member?: (E, %) -> Boolean
from HomogeneousAggregate E
- members: % -> List E
from HomogeneousAggregate E
- min: % -> E if E has OrderedSet
from HomogeneousAggregate E
- minColIndex: % -> Integer
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- minordet: % -> E if E has CommutativeRing
from MatrixCategory(E, WSVector E, WSVector E)
- minors: % -> %
minors(m)
returns the minors ofm
.
- minors: (%, WSInteger) -> %
minors(m,i)
returns thei
-th minors ofm
.
- minRowIndex: % -> Integer
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- more?: (%, NonNegativeInteger) -> Boolean
from Aggregate
- mutable?: % -> Boolean
from JLObjectType
- ncols: % -> NonNegativeInteger
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- new: (NonNegativeInteger, NonNegativeInteger, E) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- norm: % -> E
norm(m)
returns the norm ofm
, i.e. the value of the maximum singular values ofm
.
- norm: (%, WSExpression) -> E
norm(m, type)
returns the type norm ofm
. For example, norm(m
,2) returns the usual norm, norm(m
, “Infinity”) or norm(m
, “Frobenius”) returns their respective norms. See documentation for information.
- nothing?: % -> Boolean
from JLObjectType
- nrows: % -> NonNegativeInteger
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- nullity: % -> NonNegativeInteger if E has IntegralDomain
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- nullSpace: % -> List WSVector E if E has IntegralDomain
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- part: (%, WSInteger) -> WSVector E
from WSAggregate WSVector E
- parts: % -> List E
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- permanent: % -> E
permanent(m)
returns the permanent ofm
.
- Pfaffian: % -> E if E has CommutativeRing
from MatrixCategory(E, WSVector E, WSVector E)
- positiveDefinite?: % -> Boolean
positiveDefinite?(m)
checks whether or notm
is positive definite.
- positivePower: (%, Integer) -> %
from MatrixCategory(E, WSVector E, WSVector E)
- prepend: (%, WSVector E) -> %
from WSAggregate WSVector E
- qelt: (%, Integer) -> WSVector E
from WSAggregate WSVector E
- qelt: (%, Integer, Integer) -> E
qelt(mat, m, n)
returns the element (m
,n
) of the matrixmat
. No checks are done at the FriCAS level.
- qnew: (NonNegativeInteger, NonNegativeInteger) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- qr: % -> Record(q: %, r: %)
qr(m)
computes theQR
decomposition of the matrixm
.
- qsetelt!: (%, Integer, Integer, E) -> E
qsetelt!(mat,n,m,elt)
returns matrixmat
with element (n
,m
) replaced by the elementelt
.- qsetelt!: (%, Integer, WSVector E) -> %
from WSAggregate WSVector E
- qsetelt: (%, Integer, Integer, E) -> %
qsetelt(mat,n,m,elt)
returns a new matrix with element (n
,m
) replaced by the elementelt
.- qsetelt: (%, Integer, WSVector E) -> %
from WSAggregate WSVector E
- randomComplexMatrix: (WSList WSComplex, WSList WSInteger) -> WSMatrix WSComplex
randomComplexMatrix(range, dims)
returns a random matrix with random complex numbers in therange
range
and dims dimensions. example{range:=jWSList(”{1+I}”)@WSLIST(WSCPLX)} example{randomComplexMatrix(range,jWSList(”{3,3}”)$WSLIST(WSINT))@WSMAT(WSCPLX)}
- randomRealMatrix: (WSList WSReal, WSList WSInteger) -> WSMatrix WSReal
randomRealMatrix(range, dims)
returns a random matrix with random numbers in therange
range
and dims dimensions. example{range := jWSList(”{-5,5}”)$WSLIST(WSREAL)} example{randomRealMatrix(range,jWSList(”{3,3}”)$WSLIST(WSINT))@WSMAT(WSREAL)}
- rank: % -> NonNegativeInteger if E has IntegralDomain
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- removeDuplicates: % -> %
from WSAggregate WSVector E
- replacePart: (%, %) -> %
from WSAggregate WSVector E
- rest: % -> %
from WSAggregate WSVector E
- reverse: % -> %
from WSAggregate WSVector E
- reverse: (%, WSInteger) -> %
from WSAggregate WSVector E
- reverse: (%, WSList WSInteger) -> %
from WSAggregate WSVector E
- riffle: (%, %) -> %
from WSAggregate WSVector E
- riffle: (%, %, %) -> %
from WSAggregate WSVector E
- row: (%, Integer) -> %
row(mat,i)
returns thei
-th row.- row: (%, Integer) -> WSVector E
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- rowEchelon: % -> % if E has EuclideanDomain
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- scalarMatrix: (NonNegativeInteger, E) -> %
from MatrixCategory(E, WSVector E, WSVector E)
- schur: % -> Record(q: %, t: %)
schur(m)
computes the Schur decomposition of of the square matrixm
.
- setColumn!: (%, Integer, WSVector E) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, Integer, Integer, E) -> E
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, Integer, List Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, Integer, List Segment Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, Integer, WSVector E) -> %
from WSAggregate WSVector E
- setelt!: (%, List Integer, Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, List Integer, List Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, List Integer, Segment Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, List Segment Integer, Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, List Segment Integer, List Segment Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, List Segment Integer, Segment Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, Segment Integer, List Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, Segment Integer, List Segment Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt!: (%, Segment Integer, Segment Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- setelt: (%, Integer, Integer, E) -> %
setelt(mat,n,m,elt)
returns a new matrix with element (n
,m
) replaced by the elementelt
.- setelt: (%, Integer, WSVector E) -> %
from WSAggregate WSVector E
- setIntersection: (%, %) -> %
from WSAggregate WSVector E
- setsubMatrix!: (%, Integer, Integer, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- size?: (%, NonNegativeInteger) -> Boolean
from Aggregate
- smaller?: (%, %) -> Boolean if E has Comparable
from Comparable
- sort: % -> %
from WSAggregate WSVector E
- sorted?: % -> Boolean
from WSAggregate WSVector E
- square?: % -> Boolean
square?(m)
returnstrue
ifm
is a square matrix (i.e. ifm
has the same number of rows as columns) andfalse
otherwise.
- squareTop: % -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- subMatrix: (%, Integer, Integer, Integer, Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- svd: % -> Record(S: %, sv: %, V: %)
svd(m)
computes the singular value decomposition of the matrixm
.
- svdvals: % -> WSList E
svdvals(m)
returns the singular values of the matrixm
.
- swapColumns!: (%, Integer, Integer) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- symmetric?: % -> Boolean
symmetric?(m)
returnstrue
if the matrixm
is square and symmetric (i.e.m[i, j] = m[j, i]
for alli
andj
) andfalse
otherwise.
- take: (%, Integer) -> %
from WSAggregate WSVector E
- take: (%, WSList WSInteger) -> %
from WSAggregate WSVector E
- toString: (%, WSExpression) -> String
toString(mat, form)
returns the string representation ofmat
withWS
language format form.
- trace: % -> E
trace(m)
returns the trace ofm
.
- transpose: % -> %
transpose(m)
returns the tranposition ofm
.- transpose: WSVector E -> %
from MatrixCategory(E, WSVector E, WSVector E)
- union: (%, %) -> %
from WSAggregate WSVector E
- vertConcat: (%, %) -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- vertConcat: List % -> %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- vertSplit: (%, List NonNegativeInteger) -> List %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- vertSplit: (%, PositiveInteger) -> List %
from TwoDimensionalArrayCategory(E, WSVector E, WSVector E)
- zero?: % -> Boolean
from MatrixOperationsCategory(E, WSVector E, WSVector E)
- zero: (NonNegativeInteger, NonNegativeInteger) -> %
from MatrixCategory(E, WSVector E, WSVector E)
Comparable if E has Comparable
Evalable E if E has Evalable E
InnerEvalable(E, E) if E has Evalable E
MatrixCategory(E, WSVector E, WSVector E)
MatrixOperationsCategory(E, WSVector E, WSVector E)
TwoDimensionalArrayCategory(E, WSVector E, WSVector E)