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23.1 Introduction to Matrices and Linear Algebra | ||
23.2 Functions and Variables for Matrices and Linear Algebra |
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23.1.1 Dot | ||
23.1.2 Vectors | ||
23.1.3 eigen |
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The operator .
represents noncommutative multiplication and scalar
product. When the operands are 1-column or 1-row matrices a
and
b
, the expression a.b
is equivalent to
sum (a[i]*b[i], i, 1, length(a))
. If a
and b
are not
complex, this is the scalar product, also called the inner product or dot
product, of a
and b
. The scalar product is defined as
conjugate(a).b
when a
and b
are complex;
innerproduct
product.
When the operands are more general matrices,
the product is the matrix product a
and b
.
The number of rows of b
must equal the number of columns of a
,
and the result has number of rows equal to the number of rows of a
and number of columns equal to the number of columns of b
.
To distinguish .
as an arithmetic operator from the decimal point in a
floating point number, it may be necessary to leave spaces on either side.
For example, 5.e3
is 5000.0
but 5 . e3
is 5
times e3
.
There are several flags which govern the simplification of expressions
involving .
, namely dot0nscsimp
,
dot0simp
,
dot1simp
,
dotassoc
,
dotconstrules
,
dotdistrib
,
dotexptsimp
,
dotident
,
dotscrules
.
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vect
is a package of functions for vector analysis. load ("vect")
loads this package, and demo ("vect")
displays a demonstration.
The vector analysis package can combine and simplify symbolic expressions including dot products and cross products, together with the gradient, divergence, curl, and Laplacian operators. The distribution of these operators over sums or products is governed by several flags, as are various other expansions, including expansion into components in any specific orthogonal coordinate systems. There are also functions for deriving the scalar or vector potential of a field.
The vect
package contains these functions:
vectorsimp
,
scalefactors
,
express
,
potential
,
vectorpotential
.
By default the vect
package does not declare the dot operator to be a
commutative operator. To get a commutative dot operator .
, the command
declare(".", commutative)
must be executed.
Categories: Vectors Share packages Package vect
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The package eigen
contains several functions devoted to the
symbolic computation of eigenvalues and eigenvectors.
Maxima loads the package automatically if one of the functions
eigenvalues
or eigenvectors
is invoked.
The package may be loaded explicitly as load ("eigen")
.
demo ("eigen")
displays a demonstration of the capabilities
of this package.
batch ("eigen")
executes the same demonstration,
but without the user prompt between successive computations.
The functions in the eigen
package are:
innerproduct
,
unitvector
,
columnvector
,
gramschmidt
,
eigenvalues
,
eigenvectors
,
uniteigenvectors
,
similaritytransform
.
Categories: Vectors Matrices Share packages Package eigen
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Appends the column(s) given by the one or more lists (or matrices) onto the matrix M.
Categories: Matrices
Appends the row(s) given by the one or more lists (or matrices) onto the matrix M.
Categories: Matrices
Returns the adjoint of the matrix M. The adjoint matrix is the transpose of the matrix of cofactors of M.
Categories: Matrices
Returns the augmented coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m. This is the coefficient matrix with a column adjoined for the constant terms in each equation (i.e., those terms not dependent upon x_1, …, x_n).
(%i1) m: [2*x - (a - 1)*y = 5*b, c + b*y + a*x = 0]$ (%i2) augcoefmatrix (m, [x, y]); [ 2 1 - a - 5 b ] (%o2) [ ] [ a b c ]
Categories: Linear equations Matrices
Returns a n
by m Cauchy matrix with the elements a[i,j]
= 1/(x_i+y_i). The second argument of cauchy_matrix
is
optional. For this case the elements of the Cauchy matrix are
a[i,j] = 1/(x_i+x_j).
Remark: In the literature the Cauchy matrix can be found defined in two forms. A second definition is a[i,j] = 1/(x_i-y_i).
Examples:
(%i1) cauchy_matrix([x1,x2],[y1,y2]); [ 1 1 ] [ ------- ------- ] [ y1 + x1 y2 + x1 ] (%o1) [ ] [ 1 1 ] [ ------- ------- ] [ y1 + x2 y2 + x2 ] (%i2) cauchy_matrix([x1,x2]); [ 1 1 ] [ ---- ------- ] [ 2 x1 x2 + x1 ] (%o2) [ ] [ 1 1 ] [ ------- ---- ] [ x2 + x1 2 x2 ]
Categories: Matrices
Returns the characteristic polynomial for the matrix M
with respect to variable x. That is,
determinant (M - diagmatrix (length (M), x))
.
(%i1) a: matrix ([3, 1], [2, 4]); [ 3 1 ] (%o1) [ ] [ 2 4 ] (%i2) expand (charpoly (a, lambda)); 2 (%o2) lambda - 7 lambda + 10 (%i3) (programmode: true, solve (%)); (%o3) [lambda = 5, lambda = 2] (%i4) matrix ([x1], [x2]); [ x1 ] (%o4) [ ] [ x2 ] (%i5) ev (a . % - lambda*%, %th(2)[1]); [ x2 - 2 x1 ] (%o5) [ ] [ 2 x1 - x2 ] (%i6) %[1, 1] = 0; (%o6) x2 - 2 x1 = 0 (%i7) x2^2 + x1^2 = 1; 2 2 (%o7) x2 + x1 = 1 (%i8) solve ([%th(2), %], [x1, x2]); 1 2 (%o8) [[x1 = - -------, x2 = - -------], sqrt(5) sqrt(5) 1 2 [x1 = -------, x2 = -------]] sqrt(5) sqrt(5)
Categories: Matrices
Returns the coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m.
(%i1) coefmatrix([2*x-(a-1)*y+5*b = 0, b*y+a*x = 3], [x,y]); [ 2 1 - a ] (%o1) [ ] [ a b ]
Categories: Linear equations Matrices
Returns the i'th column of the matrix M. The return value is a matrix.
Categories: Matrices
Returns a matrix of one column and length (L)
rows,
containing the elements of the list L.
covect
is a synonym for columnvector
.
load ("eigen")
loads this function.
This is useful if you want to use parts of the outputs of the functions in this package in matrix calculations.
Example:
(%i1) load ("eigen")$ Warning - you are redefining the Macsyma function eigenvalues Warning - you are redefining the Macsyma function eigenvectors (%i2) columnvector ([aa, bb, cc, dd]); [ aa ] [ ] [ bb ] (%o2) [ ] [ cc ] [ ] [ dd ]
Categories: Matrices
Returns a copy of the matrix M. This is the only way to make a copy aside from copying M element by element.
Note that an assignment of one matrix to another, as in m2: m1
, does not
copy m1
. An assignment m2 [i,j]: x
or setelmx(x, i, j, m2)
also modifies m1 [i,j]
. Creating a copy with copymatrix
and then
using assignment creates a separate, modified copy.
Categories: Matrices
Computes the determinant of M by a method similar to Gaussian elimination.
The form of the result depends upon the setting of the switch ratmx
.
There is a special routine for computing sparse determinants which is called
when the switches ratmx
and sparse
Categories: Matrices
Default value: false
When detout
is true
, the determinant of a
matrix whose inverse is computed is factored out of the inverse.
For this switch to have an effect doallmxops
doscmxops
be false
(see their descriptions). Alternatively this switch can be
given to ev
Example:
(%i1) m: matrix ([a, b], [c, d]); [ a b ] (%o1) [ ] [ c d ] (%i2) detout: true$ (%i3) doallmxops: false$ (%i4) doscmxops: false$ (%i5) invert (m); [ d - b ] [ ] [ - c a ] (%o5) ------------ a d - b c
Categories: Matrices Evaluation flags
Returns a diagonal matrix of size n by n with the diagonal elements
all equal to x. diagmatrix (n, 1)
returns an identity matrix
(same as ident (n)
).
n must evaluate to an integer, otherwise diagmatrix
complains with
an error message.
x can be any kind of expression, including another matrix. If x is a matrix, it is not copied; all diagonal elements refer to the same instance, x.
Categories: Matrices
Default value: true
When doallmxops
is true
,
all operations relating to matrices are carried out.
When it is false
then the setting of the
individual dot
switches govern which operations are performed.
Categories: Matrices
Default value: true
When domxexpt
is true
,
a matrix exponential, exp (M)
where M is a matrix, is
interpreted as a matrix with element [i,j]
equal to exp (m[i,j])
.
Otherwise exp (M)
evaluates to exp (ev(M))
.
domxexpt
affects all expressions of the form
base^power
where base is an expression assumed scalar
or constant, and power is a list or matrix.
Example:
(%i1) m: matrix ([1, %i], [a+b, %pi]); [ 1 %i ] (%o1) [ ] [ b + a %pi ] (%i2) domxexpt: false$ (%i3) (1 - c)^m; [ 1 %i ] [ ] [ b + a %pi ] (%o3) (1 - c) (%i4) domxexpt: true$ (%i5) (1 - c)^m; [ %i ] [ 1 - c (1 - c) ] (%o5) [ ] [ b + a %pi ] [ (1 - c) (1 - c) ]
Categories: Matrices
Default value: true
When domxmxops
is true
, all matrix-matrix or
matrix-list operations are carried out (but not scalar-matrix
operations); if this switch is false
such operations are not carried out.
Categories: Matrices
Default value: false
When domxnctimes
is true
, non-commutative products of
matrices are carried out.
Categories: Matrices
Default value: []
dontfactor
may be set to a list of variables with respect to which
factoring is not to occur. (The list is initially empty.) Factoring also will
not take place with respect to any variables which are less important, according
the variable ordering assumed for canonical rational expression (CRE) form, than
those on the dontfactor
list.
Categories: Expressions
Default value: false
When doscmxops
is true
, scalar-matrix operations are
carried out.
Categories: Matrices
Default value: false
When doscmxplus
is true
, scalar-matrix operations yield
a matrix result. This switch is not subsumed under doallmxops
.
Categories: Matrices
Default value: true
When dot0nscsimp
is true
, a non-commutative product of zero
and a nonscalar term is simplified to a commutative product.
Categories: Simplification flags and variables
Default value: true
When dot0simp
is true
,
a non-commutative product of zero and
a scalar term is simplified to a commutative product.
Categories: Simplification flags and variables
Default value: true
When dot1simp
is true
,
a non-commutative product of one and
another term is simplified to a commutative product.
Categories: Simplification flags and variables
Default value: true
When dotassoc
is true
, an expression (A.B).C
simplifies to
A.(B.C)
.
Categories: Simplification flags and variables
Default value: true
When dotconstrules
is true
, a non-commutative product of a
constant and another term is simplified to a commutative product.
Turning on this flag effectively turns on dot0simp
,
dot0nscsimp
,
dot1simp
Categories: Simplification flags and variables
Default value: false
When dotdistrib
is true
, an expression A.(B + C)
simplifies
to A.B + A.C
.
Categories: Simplification flags and variables
Default value: true
When dotexptsimp
is true
, an expression A.A
simplifies to
A^^2
.
Categories: Simplification flags and variables
Default value: 1
dotident
is the value returned by X^^0
.
Categories: Simplification flags and variables
Default value: false
When dotscrules
is true
, an expression A.SC
or SC.A
simplifies to SC*A
and A.(SC*B)
simplifies to SC*(A.B)
.
Categories: Simplification flags and variables
Returns the echelon form of the matrix M, as produced by Gaussian elimination. The echelon form is computed from M by elementary row operations such that the first non-zero element in each row in the resulting matrix is one and the column elements under the first one in each row are all zero.
triangularize
normalize the leading non-zero element in each row.
lu_factor
cholesky
triangularized matrices.
(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]); [ 3 7 aa bb ] [ ] (%o1) [ - 1 8 5 2 ] [ ] [ 9 2 11 4 ] (%i2) echelon (M); [ 1 - 8 - 5 - 2 ] [ ] [ 28 11 ] [ 0 1 -- -- ] (%o2) [ 37 37 ] [ ] [ 37 bb - 119 ] [ 0 0 1 ----------- ] [ 37 aa - 313 ]
Categories: Linear equations Matrices
Returns a list of two lists containing the eigenvalues of the matrix M. The first sublist of the return value is the list of eigenvalues of the matrix, and the second sublist is the list of the multiplicities of the eigenvalues in the corresponding order.
eivals
is a synonym for eigenvalues
.
eigenvalues
calls the function solve
characteristic polynomial of the matrix. Sometimes solve
may not be able
to find the roots of the polynomial; in that case some other functions in this
package (except innerproduct
,
unitvector
,
columnvector
gramschmidt
In some cases the eigenvalues found by solve
may be complicated
expressions. (This may happen when solve
returns a not-so-obviously real
expression for an eigenvalue which is known to be real.) It may be possible to
simplify the eigenvalues using some other functions.
The package eigen.mac
is loaded automatically when
eigenvalues
or eigenvectors
If eigen.mac
is not already loaded,
load ("eigen")
loads it.
After loading, all functions and variables in the package are available.
Categories: Package eigen
Computes eigenvectors of the matrix M. The return value is a list of two elements. The first is a list of the eigenvalues of M and a list of the multiplicities of the eigenvalues. The second is a list of lists of eigenvectors. There is one list of eigenvectors for each eigenvalue. There may be one or more eigenvectors in each list.
eivects
is a synonym for eigenvectors
.
The package eigen.mac
is loaded automatically when
eigenvalues
If eigen.mac
is not already loaded,
load ("eigen")
loads it.
After loading, all functions and variables in the package are available.
The flags that affect this function are:
nondiagonalizable
is set to true
or false
depending on
whether the matrix is nondiagonalizable or diagonalizable after
eigenvectors
returns.
hermitianmatrix
when true
, causes the degenerate
eigenvectors of the Hermitian matrix to be orthogonalized using the
Gram-Schmidt algorithm.
knowneigvals
when true
causes the eigen
package to assume
the eigenvalues of the matrix are known to the user and stored under the global
name listeigvals
. listeigvals
should be set to a list similar
to the output eigenvalues
.
The function algsys
Sometimes if the eigenvalues are messy, algsys
may not be able to find a
solution. In some cases, it may be possible to simplify the eigenvalues by
first finding them using eigenvalues
command and then using other
functions to reduce them to something simpler. Following simplification,
eigenvectors
can be called again with the knowneigvals
flag set
to true
.
See also eigenvalues
.
Examples:
A matrix which has just one eigenvector per eigenvalue.
(%i1) M1 : matrix ([11, -1], [1, 7]); [ 11 - 1 ] (%o1) [ ] [ 1 7 ] (%i2) [vals, vecs] : eigenvectors (M1); (%o2) [[[9 - sqrt(3), sqrt(3) + 9], [1, 1]], [[[1, sqrt(3) + 2]], [[1, 2 - sqrt(3)]]]] (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i], mult[i] = vals[2][i], vec[i] = vecs[i]); val = 9 - sqrt(3) 1 mult = 1 1 vec = [[1, sqrt(3) + 2]] 1 val = sqrt(3) + 9 2 mult = 1 2 vec = [[1, 2 - sqrt(3)]] 2 (%o3) done
A matrix which has two eigenvectors for one eigenvalue (namely 2).
(%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]); [ 0 1 0 0 ] [ ] [ 0 0 0 0 ] (%o1) [ ] [ 0 0 2 0 ] [ ] [ 0 0 0 2 ] (%i2) [vals, vecs] : eigenvectors (M1); (%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]], [[0, 0, 1, 0], [0, 0, 0, 1]]]] (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i], mult[i] = vals[2][i], vec[i] = vecs[i]); val = 0 1 mult = 2 1 vec = [[1, 0, 0, 0]] 1 val = 2 2 mult = 2 2 vec = [[0, 0, 1, 0], [0, 0, 0, 1]] 2 (%o3) done
Categories: Package eigen
Returns an m by n matrix, all elements of which
are zero except for the [i, j]
element which is x.
Categories: Matrices
Returns an m by n matrix, reading the elements interactively.
If n is equal to m, Maxima prompts for the type of the matrix
(diagonal, symmetric, antisymmetric, or general) and for each element.
Each response is terminated by a semicolon ;
or dollar sign $
.
If n is not equal to m, Maxima prompts for each element.
The elements may be any expressions, which are evaluated.
entermatrix
evaluates its arguments.
(%i1) n: 3$ (%i2) m: entermatrix (n, n)$ Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General Answer 1, 2, 3 or 4 : 1$ Row 1 Column 1: (a+b)^n$ Row 2 Column 2: (a+b)^(n+1)$ Row 3 Column 3: (a+b)^(n+2)$ Matrix entered. (%i3) m; [ 3 ] [ (b + a) 0 0 ] [ ] (%o3) [ 4 ] [ 0 (b + a) 0 ] [ ] [ 5 ] [ 0 0 (b + a) ]
Categories: Console interaction Matrices
Returns a matrix generated from a, taking element
a[i_1, j_1]
as the upper-left element and
a[i_2, j_2]
as the lower-right element of the matrix.
Here a is a declared array (created by array
but not by
make_array
expression of two arguments. (An array function is created like other functions
with :=
define
,
brackets instead of parentheses.)
If j_1 is omitted, it is assumed equal to i_1. If both j_1 and i_1 are omitted, both are assumed equal to 1.
If a selected element i,j
of the array is undefined,
the matrix will contain a symbolic element a[i,j]
.
Examples:
(%i1) h [i, j] := 1 / (i + j - 1); 1 (%o1) h := --------- i, j i + j - 1 (%i2) genmatrix (h, 3, 3); [ 1 1 ] [ 1 - - ] [ 2 3 ] [ ] [ 1 1 1 ] (%o2) [ - - - ] [ 2 3 4 ] [ ] [ 1 1 1 ] [ - - - ] [ 3 4 5 ] (%i3) array (a, fixnum, 2, 2); (%o3) a (%i4) a [1, 1] : %e; (%o4) %e (%i5) a [2, 2] : %pi; (%o5) %pi (%i6) genmatrix (a, 2, 2); [ %e 0 ] (%o6) [ ] [ 0 %pi ] (%i7) genmatrix (lambda ([i, j], j - i), 3, 3); [ 0 1 2 ] [ ] (%o7) [ - 1 0 1 ] [ ] [ - 2 - 1 0 ] (%i8) genmatrix (B, 2, 2); [ B B ] [ 1, 1 1, 2 ] (%o8) [ ] [ B B ] [ 2, 1 2, 2 ]
Categories: Matrices
Carries out the Gram-Schmidt orthogonalization algorithm on x, which is
either a matrix or a list of lists. x is not modified by
gramschmidt
. The inner product employed by gramschmidt
is
F, if present, otherwise the inner product is the function
innerproduct
.
If x is a matrix, the algorithm is applied to the rows of x. If x is a list of lists, the algorithm is applied to the sublists, which must have equal numbers of elements. In either case, the return value is a list of lists, the sublists of which are orthogonal and span the same space as x. If the dimension of the span of x is less than the number of rows or sublists, some sublists of the return value are zero.
factor
results. As a consequence, the return value may contain factored integers.
load(eigen)
loads this function.
Example:
Gram-Schmidt algorithm using default inner product function.
(%i1) load (eigen)$ (%i2) x: matrix ([1, 2, 3], [9, 18, 30], [12, 48, 60]); [ 1 2 3 ] [ ] (%o2) [ 9 18 30 ] [ ] [ 12 48 60 ] (%i3) y: gramschmidt (x); 2 2 4 3 3 3 3 5 2 3 2 3 (%o3) [[1, 2, 3], [- ---, - --, ---], [- ----, ----, 0]] 2 7 7 2 7 5 5 (%i4) map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]); (%o4) [0, 0, 0]
Gram-Schmidt algorithm using a specified inner product function.
(%i1) load (eigen)$ (%i2) ip (f, g) := integrate (f * g, u, a, b); (%o2) ip(f, g) := integrate(f g, u, a, b) (%i3) y : gramschmidt([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2; %pi cos(u) - 2 (%o3) [1, sin(u), --------------] %pi (%i4) map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]), a= -%pi/2, b=%pi/2; (%o4) [0, 0, 0]
Categories: Package eigen
Returns an n by n identity matrix.
Categories: Matrices
Returns the inner product (also called the scalar product or dot product) of
x and y, which are lists of equal length, or both 1-column or 1-row
matrices of equal length. The return value is conjugate (x) . y
,
where .
is the noncommutative multiplication operator.
load ("eigen")
loads this function.
inprod
is a synonym for innerproduct
.
Categories: Package eigen
Returns the inverse of the matrix M. The inverse is computed via the LU decomposition.
When ratmx
elements of M are converted to canonical rational expressions (CRE),
and the elements of the return value are also CRE.
When ratmx
elements of M are not converted to a common representation.
In particular, float and bigfloat elements are not converted to rationals.
When detout
The global flags doallmxops
doscmxops
to prevent the determinant from being absorbed into the inverse.
xthru
invert
apart from the default arithmetic simplifications.
ratsimp
expand
In particular, when M has polynomial elements,
expand(invert(M))
might be preferable.
invert(M)
is equivalent to M^^-1
.
Categories: Matrices
Returns a list containing the elements of the matrix M.
Example:
(%i1) list_matrix_entries(matrix([a,b],[c,d])); (%o1) [a, b, c, d]
Categories: Matrices
Default value: [
lmxchar
is the character displayed as the left delimiter of a matrix.
See also rmxchar
Example:
(%i1) lmxchar: "|"$ (%i2) matrix ([a, b, c], [d, e, f], [g, h, i]); | a b c ] | ] (%o2) | d e f ] | ] | g h i ]
Categories: Display flags and variables Matrices
Returns a rectangular matrix which has the rows row_1, …, row_n. Each row is a list of expressions. All rows must be the same length.
The operations +
(addition), -
(subtraction), *
(multiplication), and /
(division), are carried out element by element
when the operands are two matrices, a scalar and a matrix, or a matrix and a
scalar. The operation ^
(exponentiation, equivalently **
)
is carried out element by element if the operands are a scalar and a matrix or
a matrix and a scalar, but not if the operands are two matrices.
All operations are normally carried out in full,
including .
(noncommutative multiplication).
Matrix multiplication is represented by the noncommutative multiplication
operator .
. The corresponding noncommutative exponentiation operator
is ^^
. For a matrix A
, A.A = A^^2
and A^^-1
is the inverse of A, if it exists.
A^^-1
is equivalent to invert(A)
.
There are switches for controlling simplification of expressions involving dot
and matrix-list operations. These are
doallmxops
,
domxexpt
,
domxmxops
,
doscmxops
,
doscmxplus
.
There are additional options which are related to matrices. These are:
lmxchar
,
rmxchar
,
ratmx
,
listarith
,
detout
,
sparse
.
There are a number of functions which take matrices as arguments or yield
matrices as return values.
See eigenvalues
,
eigenvectors
,
determinant
,
charpoly
,
genmatrix
,
addcol
,
addrow
,
copymatrix
,
transpose
,
echelon
,
rank
.
Examples:
(%i1) x: matrix ([17, 3], [-8, 11]); [ 17 3 ] (%o1) [ ] [ - 8 11 ] (%i2) y: matrix ([%pi, %e], [a, b]); [ %pi %e ] (%o2) [ ] [ a b ]
(%i3) x + y; [ %pi + 17 %e + 3 ] (%o3) [ ] [ a - 8 b + 11 ]
(%i4) x - y; [ 17 - %pi 3 - %e ] (%o4) [ ] [ - a - 8 11 - b ]
(%i5) x * y; [ 17 %pi 3 %e ] (%o5) [ ] [ - 8 a 11 b ]
(%i6) x / y; [ 17 - 1 ] [ --- 3 %e ] [ %pi ] (%o6) [ ] [ 8 11 ] [ - - -- ] [ a b ]
(%i7) x ^ 3; [ 4913 27 ] (%o7) [ ] [ - 512 1331 ]
(%i8) exp(y); [ %pi %e ] [ %e %e ] (%o8) [ ] [ a b ] [ %e %e ]
(%i9) x ^ y; [ %pi %e ] [ ] [ a b ] [ 17 3 ] (%o9) [ ] [ - 8 11 ]
(%i10) x . y; [ 3 a + 17 %pi 3 b + 17 %e ] (%o10) [ ] [ 11 a - 8 %pi 11 b - 8 %e ] (%i11) y . x; [ 17 %pi - 8 %e 3 %pi + 11 %e ] (%o11) [ ] [ 17 a - 8 b 11 b + 3 a ]
b^^m
is the same as b^m
.
(%i12) x ^^ 3; [ 3833 1719 ] (%o12) [ ] [ - 4584 395 ] (%i13) %e ^^ y; [ %pi %e ] [ %e %e ] (%o13) [ ] [ a b ] [ %e %e ]
(%i14) x ^^ -1; [ 11 3 ] [ --- - --- ] [ 211 211 ] (%o14) [ ] [ 8 17 ] [ --- --- ] [ 211 211 ] (%i15) x . (x ^^ -1); [ 1 0 ] (%o15) [ ] [ 0 1 ]
Categories: Matrices
Returns a matrix with element i,j
equal to f(M[i,j])
.
See also map
,
fullmap
,
fullmapl
,
apply
.
Categories: Matrices
Returns true
if expr is a matrix, otherwise false
.
Categories: Predicate functions Matrices
Default value: +
matrix_element_add
is the operation
invoked in place of addition in a matrix multiplication.
matrix_element_add
can be assigned any n-ary operator
(that is, a function which handles any number of arguments).
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function,
or a lambda expression.
See also matrix_element_mult
matrix_element_transpose
.
Example:
(%i1) matrix_element_add: "*"$ (%i2) matrix_element_mult: "^"$ (%i3) aa: matrix ([a, b, c], [d, e, f]); [ a b c ] (%o3) [ ] [ d e f ] (%i4) bb: matrix ([u, v, w], [x, y, z]); [ u v w ] (%o4) [ ] [ x y z ] (%i5) aa . transpose (bb); [ u v w x y z ] [ a b c a b c ] (%o5) [ ] [ u v w x y z ] [ d e f d e f ]
Categories: Matrices
Default value: *
matrix_element_mult
is the operation
invoked in place of multiplication in a matrix multiplication.
matrix_element_mult
can be assigned any binary operator.
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function,
or a lambda expression.
The dot operator .
is a useful choice in some contexts.
See also matrix_element_add
matrix_element_transpose
.
Example:
(%i1) matrix_element_add: lambda ([[x]], sqrt (apply ("+", x)))$ (%i2) matrix_element_mult: lambda ([x, y], (x - y)^2)$ (%i3) [a, b, c] . [x, y, z]; 2 2 2 (%o3) sqrt((c - z) + (b - y) + (a - x) ) (%i4) aa: matrix ([a, b, c], [d, e, f]); [ a b c ] (%o4) [ ] [ d e f ] (%i5) bb: matrix ([u, v, w], [x, y, z]); [ u v w ] (%o5) [ ] [ x y z ] (%i6) aa . transpose (bb); [ 2 2 2 ] [ sqrt((c - w) + (b - v) + (a - u) ) ] (%o6) Col 1 = [ ] [ 2 2 2 ] [ sqrt((f - w) + (e - v) + (d - u) ) ] [ 2 2 2 ] [ sqrt((c - z) + (b - y) + (a - x) ) ] Col 2 = [ ] [ 2 2 2 ] [ sqrt((f - z) + (e - y) + (d - x) ) ]
Categories: Matrices
Default value: false
matrix_element_transpose
is the operation
applied to each element of a matrix when it is transposed.
matrix_element_mult
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function, or a lambda expression.
When matrix_element_transpose
equals transpose
,
the transpose
function is applied to every element.
When matrix_element_transpose
equals nonscalars
,
the transpose
function is applied to every nonscalar element.
If some element is an atom, the nonscalars
option applies
transpose
only if the atom is declared nonscalar,
while the transpose
option always applies transpose
.
The default value, false
, means no operation is applied.
See also matrix_element_add
matrix_element_mult
.
Examples:
(%i1) declare (a, nonscalar)$ (%i2) transpose ([a, b]); [ transpose(a) ] (%o2) [ ] [ b ] (%i3) matrix_element_transpose: nonscalars$ (%i4) transpose ([a, b]); [ transpose(a) ] (%o4) [ ] [ b ] (%i5) matrix_element_transpose: transpose$ (%i6) transpose ([a, b]); [ transpose(a) ] (%o6) [ ] [ transpose(b) ] (%i7) matrix_element_transpose: lambda ([x], realpart(x) - %i*imagpart(x))$ (%i8) m: matrix ([1 + 5*%i, 3 - 2*%i], [7*%i, 11]); [ 5 %i + 1 3 - 2 %i ] (%o8) [ ] [ 7 %i 11 ] (%i9) transpose (m); [ 1 - 5 %i - 7 %i ] (%o9) [ ] [ 2 %i + 3 11 ]
Categories: Matrices
Returns the trace (that is, the sum of the elements on the main diagonal) of the square matrix M.
mattrace
is called by ncharpoly
,
charpoly
.
load ("nchrpl")
loads this function.
Categories: Matrices Package nchrpl
Returns the i, j minor of the matrix M. That is, M with row i and column j removed.
Categories: Matrices
Returns the characteristic polynomial of the matrix M
with respect to x. This is an alternative to Maxima's charpoly
.
ncharpoly
works by computing traces of powers of the given matrix,
which are known to be equal to sums of powers of the roots of the
characteristic polynomial. From these quantities the symmetric
functions of the roots can be calculated, which are nothing more than
the coefficients of the characteristic polynomial. charpoly
works by
forming the determinant of x * ident [n] - a
. Thus
ncharpoly
wins, for example, in the case of large dense matrices filled
with integers, since it avoids polynomial arithmetic altogether.
load ("nchrpl")
loads this file.
Categories: Matrices Package nchrpl
Computes the determinant of the matrix M by the Johnson-Gentleman tree
minor algorithm. newdet
returns the result in CRE form.
Categories: Matrices
Computes the permanent of the matrix M by the Johnson-Gentleman tree
minor algorithm. A permanent is like a determinant but with no sign changes.
permanent
returns the result in CRE form.
See also newdet
.
Categories: Matrices
Computes the rank of the matrix M. That is, the order of the largest non-singular subdeterminant of M.
rank may return the wrong answer if it cannot determine that a matrix element that is equivalent to zero is indeed so.
Categories: Matrices
Default value: false
When ratmx
is false
, determinant and matrix
addition, subtraction, and multiplication are performed in the
representation of the matrix elements and cause the result of
matrix inversion to be left in general representation.
When ratmx
is true
,
the 4 operations mentioned above are performed in CRE form and the
result of matrix inverse is in CRE form. Note that this may
cause the elements to be expanded (depending on the setting of ratfac
which might not always be desired.
Categories: Matrices Rational expressions
Returns the i'th row of the matrix M. The return value is a matrix.
Categories: Matrices
Default value: ]
rmxchar
is the character drawn on the right-hand side of a matrix.
See also lmxchar
.
Categories: Display flags and variables
Default value: true
When scalarmatrixp
is true
, then whenever a 1 x 1 matrix
is produced as a result of computing the dot product of matrices it
is simplified to a scalar, namely the sole element of the matrix.
When scalarmatrixp
is all
,
then all 1 x 1 matrices are simplified to scalars.
When scalarmatrixp
is false
, 1 x 1 matrices are not simplified
to scalars.
Categories: Matrices Simplification flags and variables
Here the argument coordinatetransform evaluates to the form
[[expression1, expression2, ...], indeterminate1, indeterminat2, ...]
,
where the variables indeterminate1, indeterminate2, etc. are the
curvilinear coordinate variables and where a set of rectangular Cartesian
components is given in terms of the curvilinear coordinates by
[expression1, expression2, ...]
. coordinates
is set to the vector
[indeterminate1, indeterminate2,...]
, and dimension
is set to the
length of this vector. SF[1], SF[2], …, SF[DIMENSION] are set to the
coordinate scale factors, and sfprod
is set to the product of these scale
factors. Initially, coordinates
is [X, Y, Z]
, dimension
is 3, and SF[1]=SF[2]=SF[3]=SFPROD=1, corresponding to 3-dimensional rectangular
Cartesian coordinates. To expand an expression into physical components in the
current coordinate system, there is a function with usage of the form
Categories: Package vect
Assigns x to the (i, j)'th element of the matrix M, and returns the altered matrix.
M [i, j]: x
has the same effect,
but returns x instead of M.
Categories: Matrices
similaritytransform
computes a similarity transform of the matrix
M
. It returns a list which is the output of the uniteigenvectors
command. In addition if the flag nondiagonalizable
is false
two
global matrices leftmatrix
and rightmatrix
are computed. These
matrices have the property that leftmatrix . M . rightmatrix
is a
diagonal matrix with the eigenvalues of M on the diagonal. If
nondiagonalizable
is true
the left and right matrices are not
computed.
If the flag hermitianmatrix
is true
then leftmatrix
is the
complex conjugate of the transpose of rightmatrix
. Otherwise
leftmatrix
is the inverse of rightmatrix
.
rightmatrix
is the matrix the columns of which are the unit
eigenvectors of M. The other flags (see eigenvalues
and
eigenvectors
) have the same effects since
similaritytransform
calls the other functions in the package in order
to be able to form rightmatrix
.
load ("eigen")
loads this function.
simtran
is a synonym for similaritytransform
.
Categories: Package eigen
Default value: false
When sparse
is true
, and if ratmx
is true
, then
determinant
will use special routines for computing sparse determinants.
Categories: Matrices
Returns a new matrix composed of the matrix M with rows i_1, …, i_m deleted, and columns j_1, …, j_n deleted.
Categories: Matrices
Returns the transpose of M.
If M is a matrix, the return value is another matrix N
such that N[i,j] = M[j,i]
.
If M is a list, the return value is a matrix N
of length (m)
rows and 1 column, such that N[i,1] = M[i]
.
Otherwise M is a symbol,
and the return value is a noun expression 'transpose (M)
.
Categories: Matrices
Returns the upper triangular form of the matrix M
,
as produced by Gaussian elimination.
The return value is the same as echelon
,
except that the leading nonzero coefficient in each row is not normalized to 1.
lu_factor
and cholesky
are other functions which yield
triangularized matrices.
(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]); [ 3 7 aa bb ] [ ] (%o1) [ - 1 8 5 2 ] [ ] [ 9 2 11 4 ] (%i2) triangularize (M); [ - 1 8 5 2 ] [ ] (%o2) [ 0 - 74 - 56 - 22 ] [ ] [ 0 0 626 - 74 aa 238 - 74 bb ]
Categories: Linear equations Matrices
Computes unit eigenvectors of the matrix M.
The return value is a list of lists, the first sublist of which is the
output of the eigenvalues
command, and the other sublists of which are
the unit eigenvectors of the matrix corresponding to those eigenvalues
respectively.
The flags mentioned in the description of the
eigenvectors
command have the same effects in this one as well.
When knowneigvects
is true
, the eigen
package assumes
that the eigenvectors of the matrix are known to the user and are
stored under the global name listeigvects
. listeigvects
should
be set to a list similar to the output of the eigenvectors
command.
If knowneigvects
is set to true
and the list of eigenvectors is
given the setting of the flag nondiagonalizable
may not be correct. If
that is the case please set it to the correct value. The author assumes that
the user knows what he is doing and will not try to diagonalize a matrix the
eigenvectors of which do not span the vector space of the appropriate dimension.
load ("eigen")
loads this function.
ueivects
is a synonym for uniteigenvectors
.
Categories: Package eigen
Returns x/norm(x); this is a unit vector in the same direction as x.
load ("eigen")
loads this function.
uvect
is a synonym for unitvector
.
Categories: Package eigen
Returns the vector potential of a given curl vector, in the current coordinate
system. potentialzeroloc
has a similar role as for potential
, but
the order of the left-hand sides of the equations must be a cyclic permutation
of the coordinate variables.
Categories: Package vect
Applies simplifications and expansions according to the following global flags:
expandall
, expanddot
, expanddotplus
, expandcross
, expandcrossplus
,
expandcrosscross
, expandgrad
, expandgradplus
, expandgradprod
,
expanddiv
, expanddivplus
, expanddivprod
, expandcurl
, expandcurlplus
,
expandcurlcurl
, expandlaplacian
, expandlaplacianplus
,
and expandlaplacianprod
.
All these flags have default value false
. The plus
suffix refers
to employing additivity or distributivity. The prod
suffix refers to the
expansion for an operand that is any kind of product.
expandcrosscross
Simplifies p ~ (q ~ r) to (p . r)*q - (p . q)*r.
expandcurlcurl
Simplifies curl curl p to grad div p + div grad p.
expandlaplaciantodivgrad
Simplifies laplacian p to div grad p.
expandcross
Enables expandcrossplus
and expandcrosscross
.
expandplus
Enables expanddotplus
, expandcrossplus
, expandgradplus
,
expanddivplus
, expandcurlplus
, and expandlaplacianplus
.
expandprod
Enables expandgradprod
, expanddivprod
, and expandlaplacianprod
.
These flags have all been declared evflag
.
Categories: Package vect Simplification functions
Default value: false
When vect_cross
is true
, it allows DIFF(X~Y,T) to work where
~ is defined in SHARE;VECT (where VECT_CROSS is set to true
, anyway.)
Categories: Package vect Differential calculus
Returns an m by n matrix, all elements of which are zero.
Categories: Matrices
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