The ToeplitzMatrix(V) function returns the Toeplitz Matrix corresponding to the values in V.

If T := ToeplitzMatrix(V), then T is an r x r Matrix, with entries T[i, j] = V[n+i-j] where r and n take the value (nops(V) + 1)/2.

The row dimension of the resulting Matrix is optional in the calling sequence. It is an error if a user-supplied value of r exceeds the default value.  In this case, it is an error if nops(V) is not odd.

If the symmetric option (sym) is included in the calling sequence as just the symbol symmetric or in the form symmetric=true, then the result is a symmetric r x r Matrix T with T[i, j] = V[j-i+1] for i <= j.  In this case, r defaults to the number of elements in V.  It is an error for a user-supplied value for r to exceed this default.  In this case, T is built with the symmetric shape.

Note:  ToeplitzMatrix(V, symmetric) and ToeplitzMatrix(V1) where V1 = [V[-1], V[-2], ..., V[1], V[2], ..., V[-1]] build the same Matrix (mathematically). The difference is that the former uses the symmetric shape.

If the compact option (cpt) is included in the calling sequence as just the symbol compact or in the form compact=true, then the result is built by using a shape function designed to minimize storage.  If the option is omitted or entered as compact=false, a full rectangular Matrix is constructed.  Generally, if space is not a consideration, the full rectangular form (the default) is more efficient.

Note:  If the compact form is selected, any datatype specification in the constructor options is silently ignored.  The datatype of any data retrieved from the constructed Matrix is determined by the data used to build the Matrix.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

This function is part of the LinearAlgebra package, and so it can be used in the form ToeplitzMatrix(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[ToeplitzMatrix](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\&colon;$

$\mathrm{T1}≔\mathrm{ToeplitzMatrix}\left(\left[a\&comma;b\&comma;c\right]\&comma;\mathrm{compact}\right)$

$\mathrm{T1}≔\left[\begin{array}{cc}b& a\\ c& b\end{array}\right]$

$\mathrm{MatrixOptions}\left(\mathrm{T1}\&comma;\mathrm{shape}\right)$

$\left[{\mathrm{Toeplitz}}_{\mathrm{false},\left[a\&comma;b\&comma;c\right]}\right]$

$V≔⟨q\&comma;r\&comma;s\&comma;t\&comma;u⟩\&colon;$

$\mathrm{T2}≔\mathrm{ToeplitzMatrix}\left(V\&comma;4\&comma;\mathrm{symmetric}\right)$

$\mathrm{T2}≔\left[\begin{array}{cccc}q& r& s& t\\ r& q& r& s\\ s& r& q& r\\ t& s& r& q\end{array}\right]$

$\mathrm{MatrixOptions}\left(\mathrm{T2}\&comma;\mathrm{shape}\right)$

$\left[\mathrm{symmetric}\right]$

$\mathrm{ToeplitzMatrix}\left(v\&comma;3\right)$

$\left[\begin{array}{ccc}{v}_3& v_2& v_1\\ v_4& v_3& v_2\\ v_5& v_4& v_3\end{array}\right]$