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58.1 Introduction to linearalgebra | ||

58.2 Functions and Variables for linearalgebra |

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`linearalgebra`

is a collection of functions for linear algebra.

Example:

(%i1) M : matrix ([1, 2], [1, 2]); [ 1 2 ] (%o1) [ ] [ 1 2 ] (%i2) nullspace (M); [ 1 ] [ ] (%o2) span([ 1 ]) [ - - ] [ 2 ] (%i3) columnspace (M); [ 1 ] (%o3) span([ ]) [ 1 ] (%i4) ptriangularize (M - z*ident(2), z); [ 1 2 - z ] (%o4) [ ] [ 2 ] [ 0 3 z - z ] (%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3); [ 1 - z 2 3 ] [ ] (%o5) [ 4 5 - z 6 ] [ ] [ 7 8 9 - z ] (%i6) MM : ptriangularize (M, z); [ 4 5 - z 6 ] [ ] [ 2 ] [ 66 z 102 z 132 ] [ 0 -- - -- + ----- + --- ] (%o6) [ 49 7 49 49 ] [ ] [ 3 2 ] [ 49 z 245 z 147 z ] [ 0 0 ----- - ------ - ----- ] [ 264 88 44 ] (%i7) algebraic : true; (%o7) true (%i8) tellrat (MM [3, 3]); 3 2 (%o8) [z - 15 z - 18 z] (%i9) MM : ratsimp (MM); [ 4 5 - z 6 ] [ ] [ 2 ] (%o9) [ 66 7 z - 102 z - 132 ] [ 0 -- - ------------------ ] [ 49 49 ] [ ] [ 0 0 0 ] (%i10) nullspace (MM); [ 1 ] [ ] [ 2 ] [ z - 14 z - 16 ] [ -------------- ] (%o10) span([ 8 ]) [ ] [ 2 ] [ z - 18 z - 12 ] [ - -------------- ] [ 12 ] (%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]); [ 1 2 3 4 ] [ ] [ 5 6 7 8 ] (%o11) [ ] [ 9 10 11 12 ] [ ] [ 13 14 15 16 ] (%i12) columnspace (M); [ 1 ] [ 2 ] [ ] [ ] [ 5 ] [ 6 ] (%o12) span([ ], [ ]) [ 9 ] [ 10 ] [ ] [ ] [ 13 ] [ 14 ] (%i13) apply ('orthogonal_complement, args (nullspace (transpose (M)))); [ 0 ] [ 1 ] [ ] [ ] [ 1 ] [ 0 ] (%o13) span([ ], [ ]) [ 2 ] [ - 1 ] [ ] [ ] [ 3 ] [ - 2 ]

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__Function:__**addmatrices***(*`f`,`M_1`, ...,`M_n`)Using the function

`f`as the addition function, return the sum of the matrices`M_1`, ...,`M_n`. The function`f`must accept any number of arguments (a Maxima nary function).Examples:

(%i1) m1 : matrix([1,2],[3,4])$ (%i2) m2 : matrix([7,8],[9,10])$ (%i3) addmatrices('max,m1,m2); (%o3) matrix([7,8],[9,10]) (%i4) addmatrices('max,m1,m2,5*m1); (%o4) matrix([7,10],[15,20])

__Function:__**blockmatrixp***(*`M`)Return true if and only if

`M`is a matrix and every entry of`M`is a matrix.

__Function:__**columnop***(*`M`,`i`,`j`,`theta`)If

`M`is a matrix, return the matrix that results from doing the column operation`C_i <- C_i -`

. If`theta`* C_j`M`doesn't have a row`i`or`j`, signal an error.

__Function:__**columnswap***(*`M`,`i`,`j`)If

`M`is a matrix, swap columns`i`and`j`. If`M`doesn't have a column`i`or`j`, signal an error.

__Function:__**columnspace***(*`M`)If

`M`is a matrix, return`span (v_1, ..., v_n)`

, where the set`{v_1, ..., v_n}`

is a basis for the column space of`M`. The span of the empty set is`{0}`

. Thus, when the column space has only one member, return`span ()`

.

__Function:__**copy***(*`e`)Return a copy of the Maxima expression

`e`. Although`e`can be any Maxima expression, the copy function is the most useful when`e`is either a list or a matrix; consider:(%i1) m : [1,[2,3]]$ (%i2) mm : m$ (%i3) mm[2][1] : x$ (%i4) m; (%o4) [1,[x,3]] (%i5) mm; (%o5) [1,[x,3]]

Let's try the same experiment, but this time let

`mm`be a copy of`m`(%i6) m : [1,[2,3]]$ (%i7) mm : copy(m)$ (%i8) mm[2][1] : x$ (%i9) m; (%o9) [1,[2,3]] (%i10) mm; (%o10) [1,[x,3]]

This time, the assignment to

`mm`does not change the value of`m`.

__Function:__**cholesky***(*`M`)__Function:__**cholesky***(*`M`,`field`)Return the Cholesky factorization of the matrix selfadjoint (or hermitian) matrix

`M`. The second argument defaults to 'generalring.' For a description of the possible values for`field`, see`lu_factor`

.

__Function:__**ctranspose***(*`M`)Return the complex conjugate transpose of the matrix

`M`. The function`ctranspose`

uses`matrix_element_transpose`

to transpose each matrix element.

__Function:__**diag_matrix***(*`d_1`,`d_2`,...,`d_n`)Return a diagonal matrix with diagonal entries

`d_1`,`d_2`, ...,`d_n`. When the diagonal entries are matrices, the zero entries of the returned matrix are zero matrices of the appropriate size; for example:(%i1) diag_matrix(diag_matrix(1,2),diag_matrix(3,4)); [ [ 1 0 ] [ 0 0 ] ] [ [ ] [ ] ] [ [ 0 2 ] [ 0 0 ] ] (%o1) [ ] [ [ 0 0 ] [ 3 0 ] ] [ [ ] [ ] ] [ [ 0 0 ] [ 0 4 ] ] (%i2) diag_matrix(p,q); [ p 0 ] (%o2) [ ] [ 0 q ]

__Function:__**dotproduct***(*`u`,`v`)Return the dotproduct of vectors

`u`and`v`. This is the same as`conjugate (transpose (`

. The arguments`u`)) .`v``u`and`v`must be column vectors.

__Function:__**eigens_by_jacobi***(*`A`)__Function:__**eigens_by_jacobi***(*`A`,`field_type`)Computes the eigenvalues and eigenvectors of

`A`by the method of Jacobi rotations.`A`must be a symmetric matrix (but it need not be positive definite nor positive semidefinite).`field_type`indicates the computational field, either`floatfield`

or`bigfloatfield`

. If`field_type`is not specified, it defaults to`floatfield`

.The elements of

`A`must be numbers or expressions which evaluate to numbers via`float`

or`bfloat`

(depending on`field_type`).Examples:

(%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]); [ 1 1 ] [ ------- ------- ] [ sqrt(2) sqrt(2) ] (%o1) [ ] [ 1 1 ] [ - ------- ------- ] [ sqrt(2) sqrt(2) ] (%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]); [ sqrt(3) 0 ] (%o2) [ ] [ 0 sqrt(5) ] (%i3) M : S . L . transpose (S); [ sqrt(5) sqrt(3) sqrt(5) sqrt(3) ] [ ------- + ------- ------- - ------- ] [ 2 2 2 2 ] (%o3) [ ] [ sqrt(5) sqrt(3) sqrt(5) sqrt(3) ] [ ------- - ------- ------- + ------- ] [ 2 2 2 2 ] (%i4) eigens_by_jacobi (M); The largest percent change was 0.1454972243679 The largest percent change was 0.0 number of sweeps: 2 number of rotations: 1 (%o4) [[1.732050807568877, 2.23606797749979], [ 0.70710678118655 0.70710678118655 ] [ ]] [ - 0.70710678118655 0.70710678118655 ] (%i5) float ([[sqrt(3), sqrt(5)], S]); (%o5) [[1.732050807568877, 2.23606797749979], [ 0.70710678118655 0.70710678118655 ] [ ]] [ - 0.70710678118655 0.70710678118655 ] (%i6) eigens_by_jacobi (M, bigfloatfield); The largest percent change was 1.454972243679028b-1 The largest percent change was 0.0b0 number of sweeps: 2 number of rotations: 1 (%o6) [[1.732050807568877b0, 2.23606797749979b0], [ 7.071067811865475b-1 7.071067811865475b-1 ] [ ]] [ - 7.071067811865475b-1 7.071067811865475b-1 ]

__Function:__**get_lu_factors***(*`x`)When

, then`x`= lu_factor (`A`)`get_lu_factors`

returns a list of the form`[P, L, U]`

, where`P`is a permutation matrix,`L`is lower triangular with ones on the diagonal, and`U`is upper triangular, and

.`A`=`P``L``U`

__Function:__**hankel***(*`col`)__Function:__**hankel***(*`col`,`row`)Return a Hankel matrix

`H`. The first column of`H`is`col`; except for the first entry, the last row of`H`is`row`. The default for`row`is the zero vector with the same length as`col`.

__Function:__**hessian***(*`f`,`x`)Returns the Hessian matrix of

`f`with respect to the list of variables`x`. The`(i, j)`

-th element of the Hessian matrix is`diff(`

.`f`,`x`[i], 1,`x`[j], 1)Examples:

(%i1) hessian (x * sin (y), [x, y]); [ 0 cos(y) ] (%o1) [ ] [ cos(y) - x sin(y) ] (%i2) depends (F, [a, b]); (%o2) [F(a, b)] (%i3) hessian (F, [a, b]); [ 2 2 ] [ d F d F ] [ --- ----- ] [ 2 da db ] [ da ] (%o3) [ ] [ 2 2 ] [ d F d F ] [ ----- --- ] [ da db 2 ] [ db ]

__Function:__**hilbert_matrix***(*`n`)Return the

`n`by`n`Hilbert matrix. When`n`isn't a positive integer, signal an error.

__Function:__**identfor***(*`M`)__Function:__**identfor***(*`M`,`fld`)Return an identity matrix that has the same shape as the matrix

`M`. The diagonal entries of the identity matrix are the multiplicative identity of the field`fld`; the default for`fld`is`generalring`.The first argument

`M`should be a square matrix or a non-matrix. When`M`is a matrix, each entry of`M`can be a square matrix - thus`M`can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.See also

`zerofor`

__Function:__**invert_by_lu***(*`M`,`(rng generalring)`)Invert a matrix

`M`by using the LU factorization. The LU factorization is done using the ring`rng`.

__Function:__**jacobian***(*`f`,`x`)Returns the Jacobian matrix of the list of functions

`f`with respect to the list of variables`x`. The`(i, j)`

-th element of the Jacobian matrix is`diff(`

.`f`[i],`x`[j])Examples:

(%i1) jacobian ([sin (u - v), sin (u * v)], [u, v]); [ cos(v - u) - cos(v - u) ] (%o1) [ ] [ v cos(u v) u cos(u v) ] (%i2) depends ([F, G], [y, z]); (%o2) [F(y, z), G(y, z)] (%i3) jacobian ([F, G], [y, z]); [ dF dF ] [ -- -- ] [ dy dz ] (%o3) [ ] [ dG dG ] [ -- -- ] [ dy dz ]

__Function:__**kronecker_product***(*`A`,`B`)Return the Kronecker product of the matrices

`A`and`B`.

__Function:__**listp***(*`e`,`p`)__Function:__**listp***(*`e`)Given an optional argument

`p`, return`true`

if`e`is a Maxima list and`p`evaluates to`true`

for every list element. When`listp`

is not given the optional argument, return`true`

if`e`is a Maxima list. In all other cases, return`false`

.

__Function:__**locate_matrix_entry***(*`M`,`r_1`,`c_1`,`r_2`,`c_2`,`f`,`rel`)The first argument must be a matrix; the arguments

`r_1`through`c_2`determine a sub-matrix of`M`that consists of rows`r_1`through`r_2`and columns`c_1`through`c_2`.Find a entry in the sub-matrix

`M`that satisfies some property. Three cases:(1)

and`rel`= 'bool`f`a predicate:Scan the sub-matrix from left to right then top to bottom, and return the index of the first entry that satisfies the predicate

`f`. If no matrix entry satisfies`f`, return`false`

.(2)

and`rel`= 'max`f`real-valued:Scan the sub-matrix looking for an entry that maximizes

`f`. Return the index of a maximizing entry.(3)

and`rel`= 'min`f`real-valued:Scan the sub-matrix looking for an entry that minimizes

`f`. Return the index of a minimizing entry.

__Function:__**lu_backsub***(*`M`,`b`)When

, then`M`= lu_factor (`A`,`field`)`lu_backsub (`

solves the linear system`M`,`b`)

.`A``x`=`b`

__Function:__**lu_factor***(*`M`,`field`)Return a list of the form

`[`

, or`LU`,`perm`,`fld`]`[`

, where`LU`,`perm`,`fld`,`lower-cnd``upper-cnd`](1) The matrix

`LU`contains the factorization of`M`in a packed form. Packed form means three things: First, the rows of`LU`are permuted according to the list`perm`. If, for example,`perm`is the list`[3,2,1]`

, the actual first row of the`LU`factorization is the third row of the matrix`LU`. Second, the lower triangular factor of m is the lower triangular part of`LU`with the diagonal entries replaced by all ones. Third, the upper triangular factor of`M`is the upper triangular part of`LU`.(2) When the field is either

`floatfield`

or`complexfield`

, the numbers`lower-cnd`and`upper-cnd`are lower and upper bounds for the infinity norm condition number of`M`. For all fields, the condition number might not be estimated; for such fields,`lu_factor`

returns a two item list. Both the lower and upper bounds can differ from their true values by arbitrarily large factors. (See also`mat_cond`

.)The argument

`M`must be a square matrix.The optional argument

`fld`must be a symbol that determines a ring or field. The pre-defined fields and rings are:(a)

`generalring`

- the ring of Maxima expressions, (b)`floatfield`

- the field of floating point numbers of the type double, (c)`complexfield`

- the field of complex floating point numbers of the type double, (d)`crering`

- the ring of Maxima CRE expressions, (e)`rationalfield`

- the field of rational numbers, (f)`runningerror`

- track the all floating point rounding errors, (g)`noncommutingring`

- the ring of Maxima expressions where multiplication is the non-commutative dot operator.When the field is

`floatfield`

,`complexfield`

, or`runningerror`

, the algorithm uses partial pivoting; for all other fields, rows are switched only when needed to avoid a zero pivot.Floating point addition arithmetic isn't associative, so the meaning of 'field' differs from the mathematical definition.

A member of the field

`runningerror`

is a two member Maxima list of the form`[x,n]`

,where`x`is a floating point number and`n`

is an integer. The relative difference between the 'true' value of`x`

and`x`

is approximately bounded by the machine epsilon times`n`

. The running error bound drops some terms that of the order the square of the machine epsilon.There is no user-interface for defining a new field. A user that is familiar with Common Lisp should be able to define a new field. To do this, a user must define functions for the arithmetic operations and functions for converting from the field representation to Maxima and back. Additionally, for ordered fields (where partial pivoting will be used), a user must define functions for the magnitude and for comparing field members. After that all that remains is to define a Common Lisp structure

`mring`

. The file`mring`

has many examples.To compute the factorization, the first task is to convert each matrix entry to a member of the indicated field. When conversion isn't possible, the factorization halts with an error message. Members of the field needn't be Maxima expressions. Members of the

`complexfield`

, for example, are Common Lisp complex numbers. Thus after computing the factorization, the matrix entries must be converted to Maxima expressions.See also

`get_lu_factors`

.Examples:

(%i1) w[i,j] := random (1.0) + %i * random (1.0); (%o1) w := random(1.) + %i random(1.) i, j (%i2) showtime : true$ Evaluation took 0.00 seconds (0.00 elapsed) (%i3) M : genmatrix (w, 100, 100)$ Evaluation took 7.40 seconds (8.23 elapsed) (%i4) lu_factor (M, complexfield)$ Evaluation took 28.71 seconds (35.00 elapsed) (%i5) lu_factor (M, generalring)$ Evaluation took 109.24 seconds (152.10 elapsed) (%i6) showtime : false$ (%i7) M : matrix ([1 - z, 3], [3, 8 - z]); [ 1 - z 3 ] (%o7) [ ] [ 3 8 - z ] (%i8) lu_factor (M, generalring); [ 1 - z 3 ] [ ] (%o8) [[ 3 9 ], [1, 2], generalring] [ ----- - z - ----- + 8 ] [ 1 - z 1 - z ] (%i9) get_lu_factors (%); [ 1 0 ] [ 1 - z 3 ] [ 1 0 ] [ ] [ ] (%o9) [[ ], [ 3 ], [ 9 ]] [ 0 1 ] [ ----- 1 ] [ 0 - z - ----- + 8 ] [ 1 - z ] [ 1 - z ] (%i10) %[1] . %[2] . %[3]; [ 1 - z 3 ] (%o10) [ ] [ 3 8 - z ]

__Function:__**mat_cond***(*`M`, 1)__Function:__**mat_cond***(*`M`, inf)Return the

`p`-norm matrix condition number of the matrix`m`. The allowed values for`p`are 1 and`inf`. This function uses the LU factorization to invert the matrix`m`. Thus the running time for`mat_cond`

is proportional to the cube of the matrix size;`lu_factor`

determines lower and upper bounds for the infinity norm condition number in time proportional to the square of the matrix size.

__Function:__**mat_norm***(*`M`, 1)__Function:__**mat_norm***(*`M`, inf)__Function:__**mat_norm***(*`M`, frobenius)Return the matrix

`p`-norm of the matrix`M`. The allowed values for`p`are 1,`inf`

, and`frobenius`

(the Frobenius matrix norm). The matrix`M`should be an unblocked matrix.

__Function:__**matrixp***(*`e`,`p`)__Function:__**matrixp***(*`e`)Given an optional argument

`p`, return`true`

if`e`is a matrix and`p`evaluates to`true`

for every matrix element. When`matrixp`

is not given an optional argument, return`true`

if`e`

is a matrix. In all other cases, return`false`

.See also

`blockmatrixp`

__Function:__**matrix_size***(*`M`)Return a two member list that gives the number of rows and columns, respectively of the matrix

`M`.

__Function:__**mat_fullunblocker***(*`M`)If

`M`is a block matrix, unblock the matrix to all levels. If`M`is a matrix, return`M`; otherwise, signal an error.

__Function:__**mat_trace***(*`M`)Return the trace of the matrix

`M`. If`M`isn't a matrix, return a noun form. When`M`is a block matrix,`mat_trace(M)`

returns the same value as does`mat_trace(mat_unblocker(m))`

.

__Function:__**mat_unblocker***(*`M`)If

`M`is a block matrix, unblock`M`one level. If`M`is a matrix,`mat_unblocker (M)`

returns`M`; otherwise, signal an error.Thus if each entry of

`M`is matrix,`mat_unblocker (M)`

returns an unblocked matrix, but if each entry of`M`is a block matrix,`mat_unblocker (M)`

returns a block matrix with one less level of blocking.If you use block matrices, most likely you'll want to set

`matrix_element_mult`

to`"."`

and`matrix_element_transpose`

to`'transpose`

. See also`mat_fullunblocker`

.Example:

(%i1) A : matrix ([1, 2], [3, 4]); [ 1 2 ] (%o1) [ ] [ 3 4 ] (%i2) B : matrix ([7, 8], [9, 10]); [ 7 8 ] (%o2) [ ] [ 9 10 ] (%i3) matrix ([A, B]); [ [ 1 2 ] [ 7 8 ] ] (%o3) [ [ ] [ ] ] [ [ 3 4 ] [ 9 10 ] ] (%i4) mat_unblocker (%); [ 1 2 7 8 ] (%o4) [ ] [ 3 4 9 10 ]

__Function:__**nullspace***(*`M`)If

`M`is a matrix, return`span (v_1, ..., v_n)`

, where the set`{v_1, ..., v_n}`

is a basis for the nullspace of`M`. The span of the empty set is`{0}`

. Thus, when the nullspace has only one member, return`span ()`

.

__Function:__**nullity***(*`M`)If

`M`is a matrix, return the dimension of the nullspace of`M`.

__Function:__**orthogonal_complement***(*`v_1`, ...,`v_n`)Return

`span (u_1, ..., u_m)`

, where the set`{u_1, ..., u_m}`

is a basis for the orthogonal complement of the set`(v_1, ..., v_n)`

.Each vector

`v_1`through`v_n`must be a column vector.

__Function:__**polynomialp***(*`p`,`L`,`coeffp`,`exponp`)__Function:__**polynomialp***(*`p`,`L`,`coeffp`)__Function:__**polynomialp***(*`p`,`L`)Return

`true`

if`p`is a polynomial in the variables in the list`L`. The predicate`coeffp`must evaluate to`true`

for each coefficient, and the predicate`exponp`must evaluate to`true`

for all exponents of the variables in`L`. If you want to use a non-default value for`exponp`, you must supply`coeffp`with a value even if you want to use the default for`coeffp`.The command

`polynomialp (`

is equivalent to`p`,`L`,`coeffp`)`polynomialp (`

and`p`,`L`,`coeffp`, 'nonnegintegerp)`polynomialp (`

is equivalent to`p`,`L`)`polynomialp (`

.`p`, L`,`'constantp, 'nonnegintegerp)The polynomial needn't be expanded:

(%i1) polynomialp ((x + 1)*(x + 2), [x]); (%o1) true (%i2) polynomialp ((x + 1)*(x + 2)^a, [x]); (%o2) false

An example using non-default values for coeffp and exponp:

(%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp); (%o1) true (%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp, numberp); (%o2) true

Polynomials with two variables:

(%i1) polynomialp (x^2 + 5*x*y + y^2, [x]); (%o1) false (%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]); (%o2) true

__Function:__**polytocompanion***(*`p`,`x`)If

`p`is a polynomial in`x`, return the companion matrix of`p`. For a monic polynomial`p`of degree`n`, we have

.`p`= (-1)^`n`charpoly (polytocompanion (`p`,`x`))When

`p`isn't a polynomial in`x`, signal an error.

__Function:__**ptriangularize***(*`M`,`v`)If

`M`is a matrix with each entry a polynomial in`v`, return a matrix`M2`such that(1)

`M2`is upper triangular,(2)

, where`M2`=`E_n`...`E_1``M``E_1`through`E_n`are elementary matrices whose entries are polynomials in`v`,(3)

`|det (`

,`M`)| = |det (`M2`)|Note: This function doesn't check that every entry is a polynomial in

`v`.

__Function:__**rowop***(*`M`,`i`,`j`,`theta`)If

`M`is a matrix, return the matrix that results from doing the row operation`R_i <- R_i - theta * R_j`

. If`M`doesn't have a row`i`or`j`, signal an error.

__Function:__**rank***(*`M`)Return the rank of that matrix

`M`. The rank is the dimension of the column space.Example:

(%i1) rank(matrix([1,2],[2,4])); (%o1) 1 (%i2) rank(matrix([1,b],[c,d])); Proviso: {d - b c # 0} (%o2) 2

__Function:__**rowswap***(*`M`,`i`,`j`)If

`M`is a matrix, swap rows`i`and`j`. If`M`doesn't have a row`i`or`j`, signal an error.

__Function:__**toeplitz***(*`col`)__Function:__**toeplitz***(*`col`,`row`)Return a Toeplitz matrix

`T`. The first first column of`T`is`col`; except for the first entry, the first row of`T`is`row`. The default for`row`is complex conjugate of`col`.Example:

(%i1) toeplitz([1,2,3],[x,y,z]); [ 1 y z ] [ ] (%o1) [ 2 1 y ] [ ] [ 3 2 1 ] (%i2) toeplitz([1,1+%i]); [ 1 1 - %I ] (%o2) [ ] [ %I + 1 1 ]

__Function:__**vandermonde_matrix***([*`x_1`, ...,`x_n`])Return a

`n`by`n`matrix whose`i`-th row is`[1,`

.`x_i`,`x_i`^2, ...`x_i`^(`n`-1)]

__Function:__**zerofor***(*`M`)__Function:__**zerofor***(*`M`,`fld`)Return a zero matrix that has the same shape as the matrix

`M`. Every entry of the zero matrix is the additive identity of the field`fld`; the default for`fld`is`generalring`.The first argument

`M`should be a square matrix or a non-matrix. When`M`is a matrix, each entry of`M`can be a square matrix - thus`M`can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.See also

`identfor`

__Function:__**zeromatrixp***(*`M`)If

`M`is not a block matrix, return`true`

if`is (equal (`

is true for each element`e`, 0))`e`of the matrix`M`. If`M`is a block matrix, return`true`

if`zeromatrixp`

evaluates to`true`

for each element of`e`.

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