[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
26.1 Introduction to Matrices and Linear Algebra | ||
26.2 Definitions for Matrices and Linear Algebra |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
26.1.1 Dot | ||
26.1.2 Vectors | ||
26.1.3 eigen |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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
in the eigen
package provides the complex scalar 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
dot
, dot0nscsimp
, dot0simp
, dot1simp
, dotassoc
,
dotconstrules
, dotdistrib
, dotexptsimp
, dotident
,
and dotscrules
.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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
, and vectorpotential
.
Warning: the vect
package declares the dot operator .
to be a commutative operator.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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
,
and similaritytransform
.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Appends the column(s) given by the one or more lists (or matrices) onto the matrix M.
Appends the row(s) given by the one or more lists (or matrices) onto the matrix M.
Returns the adjoint of the matrix M. The adjoint matrix is the transpose of the matrix of cofactors of M.
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 ] |
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) |
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 ] |
Returns the i'th column of the matrix M. The return value is a matrix.
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 ] |
Returns the complex conjugate of x.
(%i1) declare ([aa, bb], real, cc, complex, ii, imaginary); (%o1) done (%i2) conjugate (aa + bb*%i); (%o2) aa - %i bb (%i3) conjugate (cc); (%o3) conjugate(cc) (%i4) conjugate (ii); (%o4) - ii (%i5) conjugate (xx + yy); (%o5) conjugate(yy) + conjugate(xx) |
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.
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
are both true
.
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
and doscmxops
should be
false
(see their descriptions). Alternatively this switch can be
given to ev
which causes the other two to be set correctly.
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 |
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.
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.
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) ] |
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.
Default value: false
When domxnctimes
is true
, non-commutative products of
matrices are carried out.
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.
Default value: false
When doscmxops
is true
, scalar-matrix operations are
carried out.
Default value: false
When doscmxplus
is true
, scalar-matrix operations yield
a matrix result. This switch is not subsumed under doallmxops
.
Default value: true
When dot0nscsimp
is true
, a non-commutative product of zero
and a nonscalar term is simplified to a commutative product.
Default value: true
When dot0simp
is true
,
a non-commutative product of zero and
a scalar term is simplified to a commutative product.
Default value: true
When dot1simp
is true
,
a non-commutative product of one and
another term is simplified to a commutative product.
Default value: true
When dotassoc
is true
, an expression (A.B).C
simplifies to
A.(B.C)
.
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
, and
dot1simp
as well.
Default value: false
When dotdistrib
is true
, an expression A.(B + C)
simplifies to A.B + A.C
.
Default value: true
When dotexptsimp
is true
, an expression A.A
simplifies to A^^2
.
Default value: 1
dotident
is the value returned by X^^0
.
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)
.
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
also carries out Gaussian elimination,
but it does not normalize the leading non-zero element in each row.
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) echelon (M); [ 1 - 8 - 5 - 2 ] [ ] [ 28 11 ] [ 0 1 -- -- ] (%o2) [ 37 37 ] [ ] [ 37 bb - 119 ] [ 0 0 1 ----------- ] [ 37 aa - 313 ] |
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
to find the roots of the
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
and
gramschmidt
) will not work.
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
is referenced.
If eigen.mac
is not already loaded,
load ("eigen")
loads it.
After loading, all functions and variables in the package are available.
takes a matrix M as its argument and returns a list
of lists the first sublist of which is the output of eigenvalues
and the other sublists of which are the eigenvectors of the
matrix corresponding to those eigenvalues respectively.
The calculated eigenvectors and the unit eigenvectors of the matrix are the
right eigenvectors and the right unit eigenvectors respectively.
eivects
is a synonym for eigenvectors
.
The package eigen.mac
is loaded automatically when
eigenvalues
or eigenvectors
is referenced.
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
is used here to solve for the eigenvectors. 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
.
Returns an m by n matrix, all elements of which
are zero except for the [i, j]
element which is x.
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) ] |
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 an array (created by array
but not by make_array
)
or an array function.
(An array function is created like other functions with :=
or define
,
but arguments are enclosed in square 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]
.
(%i1) h[i,j] := 1/(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)$ (%i4) a[1,1]: %e$ (%i5) a[2,2]: %pi$ (%i6) kill (a[1,2], a[2,1])$ (%i7) genmatrix (a, 2, 2); [ %e a ] [ 1, 2 ] (%o7) [ ] [ a %pi ] [ 2, 1 ] |
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
.
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
is called at each stage of the algorithm to simplify intermediate results.
As a consequence, the return value may contain factored integers.
gschmit
(note spelling) is a synonym for gramschmidt
.
load ("eigen")
loads this function.
Example:
(%i1) load ("eigen")$ Warning - you are redefining the Macsyma function eigenvalues Warning - you are redefining the Macsyma function eigenvectors (%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) i: innerproduct$ (%i5) [i (y[1], y[2]), i (y[2], y[3]), i (y[3], y[1])]; (%o5) [0, 0, 0] |
hach
is an implementation of Hacijan's linear programming algorithm.
load ("kach")
loads this function.
demo ("kach")
executes a demonstration of this function.
Returns an n by n identity matrix.
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
.
Returns the inverse of the matrix M. The inverse is computed by the adjoint method.
This allows a user to compute the inverse of a matrix with bfloat entries or polynomials with floating pt. coefficients without converting to cre-form.
Cofactors are computed by the determinant
function,
so if ratmx
is false
the inverse is computed
without changing the representation of the elements.
The current implementation is inefficient for matrices of high order.
When detout
is true
, the determinant is factored out of the
inverse.
The elements of the inverse are not automatically expanded.
If M has polynomial elements, better appearing output can be
generated by expand (invert (m)), detout
.
If it is desirable to then
divide through by the determinant this can be accomplished by xthru (%)
or alternatively from scratch by
expand (adjoint (m)) / expand (determinant (m)) invert (m) := adjoint (m) / determinant (m) |
See ^^
(noncommutative exponent) for another method of inverting a matrix.
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 ] |
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.
There are switches for controlling simplification of expressions
involving dot and matrix-list operations.
These are
doallmxops
, domxexpt
domxmxops
, doscmxops
, and doscmxplus
.
There are additional options which are related to matrices. These are:
lmxchar
, rmxchar
, ratmx
, listarith
, detout
,
scalarmatrix
,
and 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
,
and 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 ] |
Returns a matrix with element i,j
equal to f(M[i,j])
.
See also map
, fullmap
, fullmapl
, and apply
.
Returns true
if expr is a matrix, otherwise false
.
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
and 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 ] |
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
and 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) ) ] |
Default value: false
matrix_element_transpose
is the operation
applied to each element of a matrix when it is transposed.
matrix_element_mult
can be assigned any unary operator.
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
and 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 ] |
Returns the trace (that is, the sum of the elements on the main diagonal) of the square matrix M.
mattrace
is called by ncharpoly
,
an alternative to Maxima's charpoly
.
load ("nchrpl")
loads this function.
Returns the i, j minor of the matrix M. That is, M with row i and column j removed.
If a non-commutative exponential expression is too
wide to be displayed as a^^b
it appears as ncexpt (a,b)
.
ncexpt
is not the name of a function or operator;
the name only appears in output, and is not recognized in input.
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.
Computes the determinant of the matrix or array M by the Johnson-Gentleman tree minor algorithm. The argument n is the order; it is optional if M is a matrix.
Makes atoms behave as does a list or matrix with respect to the dot operator.
Returns true
if expr is a non-scalar, i.e., it contains
atoms declared as non-scalars, lists, or matrices.
Computes the permanent of the matrix M. A permanent is like a determinant but with no sign changes.
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.
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.
Returns the i'th row of the matrix M. The return value is a matrix.
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.
Here coordinatetransform
evaluates to the form [[expression1, expression2, ...],
indeterminate1, indeterminat2, ...], where 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
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.
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
.
Default value: false
When sparse
is true
, and if ratmx
is true
, then determinant
will use special routines for computing sparse determinants.
Returns a new matrix composed of the matrix M with rows i_1, ..., i_m deleted, and columns j_1, ..., j_n deleted.
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]
.
Otherwise M is a list, and the return value is a matrix N
of length (m)
rows and 1 column, such that N[i,1] = M[i]
.
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 ] |
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
.
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
.
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
.
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.)
Returns an m by n matrix, all elements of which are zero.
[
and ]
mark the beginning and end, respectively, of a list.
[
and ]
also enclose the subscripts of
a list, array, hash array, or array function.
Examples:
(%i1) x: [a, b, c]; (%o1) [a, b, c] (%i2) x[3]; (%o2) c (%i3) array (y, fixnum, 3); (%o3) y (%i4) y[2]: %pi; (%o4) %pi (%i5) y[2]; (%o5) %pi (%i6) z['foo]: 'bar; (%o6) bar (%i7) z['foo]; (%o7) bar (%i8) g[k] := 1/(k^2+1); 1 (%o8) g := ------ k 2 k + 1 (%i9) g[10]; 1 (%o9) --- 101 |
[ << ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This document was generated on March, 19 2006 using texi2html 1.76.