1.6 Expanding to Higher Dimensions

To get higher dimensional aggregates, you can create one-dimensional aggregates with elements that are themselves aggregates, for example, lists of lists, one-dimensional arrays of lists of multisets, and so on. For applications requiring two-dimensional homogeneous aggregates, you will likely find two-dimensional arrays matrix and matrices most useful.

The entries in TwoDimensionalArray and Matrix objects are all the same type, except that those for Matrix must belong to a Ring. You create and access elements in roughly the same way. Since matrices have an understood algebraic structure, certain algebraic operations are available for matrices but not for arrays. Because of this, we limit our discussion here to Matrix, that can be regarded as an extension of TwoDimensionalArray. See TwoDimensionalArray for more information about arrays. For more information about FriCAS’s linear algebra facilities, see Matrix, Permanent, SquareMatrix, Vector, see Section ugProblemEigen (computation of eigenvalues and eigenvectors), and Section ugProblemLinPolEqn (solution of linear and polynomial equations).

You can create a matrix from a list of lists, matrix:creating where each of the inner lists represents a row of the matrix.

m := matrix([ [1,2], [3,4] ])

\[\begin{split}\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]\end{split}\]

_{Type: Matrix Integer}

The collections construct (see ugLangIts) is useful for creating matrices whose entries are given by formulas.

matrix([ [1/(i + j - x) for i in 1..4] for j in 1..4])

\[\begin{split}\left[ \begin{array}{cccc} -{1 \over {x -2}} & -{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} \\ -{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} \\ -{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} \\ -{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} & -{1 \over {x -8}} \end{array} \right]\end{split}\]

_{Type: Matrix Fraction Polynomial Integer}

Let vm denote the three by three Vandermonde matrix.

vm := matrix [ [1,1,1], [x,y,z], [x*x,y*y,z*z] ]

\[\begin{split}\left[ \begin{array}{ccc} 1 & 1 & 1 \\ x & y & z \\ {{x} ^ {2}} & {{y} ^ {2}} & {{z} ^ {2}} \end{array} \right]\end{split}\]

_{Type: Matrix Polynomial Integer}

Use this syntax to extract an entry in the matrix.

vm(3,3)

\[{z} ^ {2}\]

_{Type: Polynomial Integer}

You can also pull out a row or a column.

column(vm,2)

\[\left[ 1, \: y, \: {{y} ^ {2}} \right]\]

_{Type: Vector Polynomial Integer}

You can do arithmetic.

vm * vm

\[\begin{split}\left[ \begin{array}{ccc} {{{x} ^ {2}}+x+1} & {{{y} ^ {2}}+y+1} & {{{z} ^ {2}}+z+1} \\ {{{{x} ^ {2}} \ z}+{x \ y}+x} & {{{{y} ^ {2}} \ z}+{{y} ^ {2}}+x} & {{{z} ^ {3}}+{y \ z}+x} \\ {{{{x} ^ {2}} \ {{z} ^ {2}}}+{x \ {{y} ^ {2}}}+{{x} ^ {2}}} & {{{{y} ^ {2}} \ {{z} ^ {2}}}+{{y} ^ {3}}+{{x} ^ {2}}} & {{{z} ^ {4}}+{{{y} ^ {2}} \ z}+{{x} ^ {2}}} \end{array} \right]\end{split}\]

_{Type: Matrix Polynomial Integer}

You can perform operations such as transpose, trace, and determinant.

factor determinant vm

\[{\left( y -x \right)} \ {\left( z -y \right)} \ {\left( z -x \right)}\]

_{Type: Factored Polynomial Integer}