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# 23. Matrices and Linear Algebra

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## 23.1 Introduction to Matrices and Linear Algebra

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### 23.1.1 Dot

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 `dot0nscsimp`, `dot0simp`, `dot1simp`, `dotassoc`, `dotconstrules`, `dotdistrib`, `dotexptsimp`, `dotident`, and `dotscrules`.

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### 23.1.2 Vectors

`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`.

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.

·

@ref{Category: Vectors} · @ref{Category: Share packages} · @ref{Category: Package vect}

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### 23.1.3 eigen

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`.

·

@ref{Category: Vectors} · @ref{Category: Matrices} · @ref{Category: Share packages} · @ref{Category: Package eigen}

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## 23.2 Functions and Variables for Matrices and Linear Algebra

Function: addcol (M, list_1, …, list_n)

  Appends the column(s) given by the one or more lists (or matrices) onto the matrix M.

  ·

  @ref{Category: Matrices}

Function: addrow (M, list_1, …, list_n)

  Appends the row(s) given by the one or more lists (or matrices) onto the matrix M.

  ·

  @ref{Category: Matrices}

Function: adjoint (M)

  Returns the adjoint of the matrix M. The adjoint matrix is the transpose of the matrix of cofactors of M.

  ·

  @ref{Category: Matrices}

Function: augcoefmatrix ([eqn_1, …, eqn_m], [x_1, …, x_n])

  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   ]

  ·

  @ref{Category: Linear equations} · @ref{Category: Matrices}

Function: cauchy_matrix ([x_1, x_2, …, x_m], [y_1, y_2, …, y_n])
Function: cauchy_matrix ([x_1, x_2, …, x_n])

  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   ]

  ·

  @ref{Category: Matrices}

Function: charpoly (M, x)

  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)

  ·

  @ref{Category: Matrices}

Function: coefmatrix ([eqn_1, …, eqn_m], [x_1, …, x_n])

  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   ]

  ·

  @ref{Category: Linear equations} · @ref{Category: Matrices}

Function: col (M, i)

  Returns the i'th column of the matrix M. The return value is a matrix.

  ·

  @ref{Category: Matrices}

Function: columnvector (L)
Function: covect (L)

  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 ]

  ·

  @ref{Category: Matrices}

Function: copymatrix (M)

  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.

  ·

  @ref{Category: Matrices}

Function: determinant (M)

  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`.

  ·

  @ref{Category: Matrices}

Option variable: detout

  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

  ·

  @ref{Category: Matrices} · @ref{Category: Evaluation flags}

Function: diagmatrix (n, x)

  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.

  ·

  @ref{Category: Matrices}

Option variable: doallmxops

  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.

  ·

  @ref{Category: Matrices}

Option variable: domxexpt

  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)    ]

  ·

  @ref{Category: Matrices}

Option variable: domxmxops

  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.

  ·

  @ref{Category: Matrices}

Option variable: domxnctimes

  Default value: `false`

  When `domxnctimes` is `true`, non-commutative products of matrices are carried out.

  ·

  @ref{Category: Matrices}

Option variable: dontfactor

  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.

  ·

  @ref{Category: Expressions}

Option variable: doscmxops

  Default value: `false`

  When `doscmxops` is `true`, scalar-matrix operations are carried out.

  ·

  @ref{Category: Matrices}

Option variable: doscmxplus

  Default value: `false`

  When `doscmxplus` is `true`, scalar-matrix operations yield a matrix result. This switch is not subsumed under `doallmxops`.

  ·

  @ref{Category: Matrices}

Option variable: dot0nscsimp

  Default value: `true`

  When `dot0nscsimp` is `true`, a non-commutative product of zero and a nonscalar term is simplified to a commutative product.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dot0simp

  Default value: `true`

  When `dot0simp` is `true`, a non-commutative product of zero and a scalar term is simplified to a commutative product.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dot1simp

  Default value: `true`

  When `dot1simp` is `true`, a non-commutative product of one and another term is simplified to a commutative product.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dotassoc

  Default value: `true`

  When `dotassoc` is `true`, an expression `(A.B).C` simplifies to `A.(B.C)`.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dotconstrules

  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.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dotdistrib

  Default value: `false`

  When `dotdistrib` is `true`, an expression `A.(B + C)` simplifies to `A.B + A.C`.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dotexptsimp

  Default value: `true`

  When `dotexptsimp` is `true`, an expression `A.A` simplifies to `A^^2`.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dotident

  Default value: 1

  `dotident` is the value returned by `X^^0`.

  ·

  @ref{Category: Simplification flags and variables}

Option variable: dotscrules

  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)`.

  ·

  @ref{Category: Simplification flags and variables}

Function: echelon (M)

  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 ]

  ·

  @ref{Category: Linear equations} · @ref{Category: Matrices}

Function: eigenvalues (M)
Function: eivals (M)

  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.

  ·

  @ref{Category: Package eigen}

Function: eigenvectors (M)
Function: eivects (M)

  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` 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`.

  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

  ·

  @ref{Category: Package eigen}

Function: ematrix (m, n, x, i, j)

  Returns an m by n matrix, all elements of which are zero except for the `[i, j]` element which is x.

  ·

  @ref{Category: Matrices}

Function: entermatrix (m, n)

  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)  ]

  ·

  @ref{Category: Console interaction} · @ref{Category: Matrices}

Function: genmatrix (a, i_2, j_2, i_1, j_1)
Function: genmatrix (a, i_2, j_2, i_1)
Function: genmatrix (a, i_2, j_2)

  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` ) or an undeclared array, or an array function, or a lambda expression of two arguments. (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]`.

  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 ]

  ·

  @ref{Category: Matrices}

Function: gramschmidt (x)
Function: gramschmidt (x, F)

  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` is called at each stage of the algorithm to simplify intermediate 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]

  ·

  @ref{Category: Package eigen}

Function: ident (n)

  Returns an n by n identity matrix.

  ·

  @ref{Category: Matrices}

Function: innerproduct (x, y)
Function: inprod (x, y)

  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`.

  ·

  @ref{Category: Package eigen}

Function: invert (M)

  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 point 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.

  ·

  @ref{Category: Matrices}

Function: list_matrix_entries (M)

  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]

  ·

  @ref{Category: Matrices}

Option variable: lmxchar

  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 ]

  ·

  @ref{Category: Display flags and variables} · @ref{Category: Matrices}

Function: matrix (row_1, …, row_n)

  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:

  - Construction of matrices from lists.

      (%i1) x: matrix ([17, 3], [-8, 11]);
                                 [ 17   3  ]
      (%o1)                      [         ]
                                 [ - 8  11 ]
      (%i2) y: matrix ([%pi, %e], [a, b]);
                                 [ %pi  %e ]
      (%o2)                      [         ]
                                 [  a   b  ]

  - Addition, element by element.

      (%i3) x + y;
                            [ %pi + 17  %e + 3 ]
      (%o3)                 [                  ]
                            [  a - 8    b + 11 ]

  - Subtraction, element by element.

      (%i4) x - y;
                            [ 17 - %pi  3 - %e ]
      (%o4)                 [                  ]
                            [ - a - 8   11 - b ]

  - Multiplication, element by element.

      (%i5) x * y;
                              [ 17 %pi  3 %e ]
      (%o5)                   [              ]
                              [ - 8 a   11 b ]

  - Division, element by element.

      (%i6) x / y;
                              [ 17       - 1 ]
                              [ ---  3 %e    ]
                              [ %pi          ]
      (%o6)                   [              ]
                              [   8    11    ]
                              [ - -    --    ]
                              [   a    b     ]

  - Matrix to a scalar exponent, element by element.

      (%i7) x ^ 3;
                               [ 4913    27  ]
      (%o7)                    [             ]
                               [ - 512  1331 ]

  - Scalar base to a matrix exponent, element by element.

      (%i8) exp(y);
                               [   %pi    %e ]
                               [ %e     %e   ]
      (%o8)                    [             ]
                               [    a     b  ]
                               [  %e    %e   ]

  - Matrix base to a matrix exponent. This is not carried out element by element.

      (%i9) x ^ y;
                                      [ %pi  %e ]
                                      [         ]
                                      [  a   b  ]
                           [ 17   3  ]
      (%o9)                [         ]
                           [ - 8  11 ]

  - Noncommutative matrix multiplication.

      (%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   ]

  - Noncommutative matrix exponentiation. A scalar base b to a matrix power M is carried out element by element and so `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   ]

  - A matrix raised to a -1 exponent with noncommutative exponentiation is the matrix inverse, if it exists.

      (%i14) x ^^ -1;
                               [ 11      3  ]
                               [ ---  - --- ]
                               [ 211    211 ]
      (%o14)                   [            ]
                               [  8    17   ]
                               [ ---   ---  ]
                               [ 211   211  ]
      (%i15) x . (x ^^ -1);
                                  [ 1  0 ]
      (%o15)                      [      ]
                                  [ 0  1 ]

  ·

  @ref{Category: Matrices}

Function: matrixmap (f, M)

  Returns a matrix with element `i,j` equal to `f(M[i,j])`.

  See also `map`, `fullmap`, `fullmapl`, and `apply`.

  ·

  @ref{Category: Matrices}

Function: matrixp (expr)

  Returns `true` if expr is a matrix, otherwise `false`.

  ·

  @ref{Category: Predicate functions} · @ref{Category: Matrices}

Option variable: matrix_element_add

  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  ]

  ·

  @ref{Category: Matrices}

Option variable: matrix_element_mult

  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) ) ]

  ·

  @ref{Category: Matrices}

Option variable: matrix_element_transpose

  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   ]

  ·

  @ref{Category: Matrices}

Function: mattrace (M)

  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.

  ·

  @ref{Category: Matrices} · @ref{Category: Package nchrpl}

Function: minor (M, i, j)

  Returns the i, j minor of the matrix M. That is, M with row i and column j removed.

  ·

  @ref{Category: Matrices}

Function: ncharpoly (M, x)

  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.

  ·

  @ref{Category: Matrices} · @ref{Category: Package nchrpl}

Function: newdet (M)

  Computes the determinant of the matrix M by the Johnson-Gentleman tree minor algorithm. `newdet` returns the result in CRE form.

  ·

  @ref{Category: Matrices}

Function: permanent (M)

  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`.

  ·

  @ref{Category: Matrices}

Function: rank (M)

  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.

  ·

  @ref{Category: Matrices}

Option variable: ratmx

  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.

  ·

  @ref{Category: Matrices} · @ref{Category: Rational expressions}

Function: row (M, i)

  Returns the i'th row of the matrix M. The return value is a matrix.

  ·

  @ref{Category: Matrices}

Option variable: rmxchar

  Default value: `]`

  `rmxchar` is the character drawn on the right-hand side of a matrix.

  See also `lmxchar`.

  ·

  @ref{Category: Display flags and variables}

Option variable: scalarmatrixp

  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.

  ·

  @ref{Category: Matrices} · @ref{Category: Simplification flags and variables}

Function: scalefactors (coordinatetransform)

  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

  ·

  @ref{Category: Package vect}

Function: setelmx (x, i, j, M)

  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.

  ·

  @ref{Category: Matrices}

Function: similaritytransform (M)
Function: simtran (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`.

  ·

  @ref{Category: Package eigen}

Option variable: sparse

  Default value: `false`

  When `sparse` is `true`, and if `ratmx` is `true`, then `determinant` will use special routines for computing sparse determinants.

  ·

  @ref{Category: Matrices}

Function: submatrix (i_1, …, i_m, M, j_1, …, j_n)
Function: submatrix (i_1, …, i_m, M)
Function: submatrix (M, j_1, …, j_n)

  Returns a new matrix composed of the matrix M with rows i_1, …, i_m deleted, and columns j_1, …, j_n deleted.

  ·

  @ref{Category: Matrices}

Function: transpose (M)

  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)`.

  ·

  @ref{Category: Matrices}

Function: triangularize (M)

  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 ]

  ·

  @ref{Category: Linear equations} · @ref{Category: Matrices}

Function: uniteigenvectors (M)
Function: ueivects (M)

  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`.

  ·

  @ref{Category: Package eigen}

Function: unitvector (x)
Function: uvect (x)

  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`.

  ·

  @ref{Category: Package eigen}

Function: vectorpotential (givencurl)

  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.

  ·

  @ref{Category: Package vect}

Function: vectorsimp (expr)

  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`.

  ·

  @ref{Category: Package vect} · @ref{Category: Simplification functions}

Option variable: vect_cross

  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.)

  ·

  @ref{Category: Package vect} · @ref{Category: Differential calculus}

Function: zeromatrix (m, n)

  Returns an m by n matrix, all elements of which are zero.

  ·

  @ref{Category: Matrices}

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