find the basis for a Jordan algebra of matrices - Maple Programming Help

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LieAlgebras[JordanMatrices] - find the basis for a Jordan algebra of matrices

LieAlgebras[JordanProduct] - find the Jordan product of two Jordan matrices

Calling Sequences

JordanMatrices(n, alg, option )

JordanProduct(A, B)

Parameters

n        - an integer

alg      - a name or string of an initialized algebra, the string "R" or the string "C"

option   - the keyword argument signature = [p, q], where p and q are integers and p + q = n

A, B     - square matrices

Description

 • Let be the algebra of real numbers, the complex numbers, the quaternions, the octonions, or one of their split versions. A Jordan matrix is a square matrix with entries in which is Hermitian with respect to the conjugation in the algebra, that is,  More generally, if is the diagonal matrix ${I}_{\mathrm{pq}}=\left[\begin{array}{rr}{I}_{p}& 0\\ 0& -{I}_{q}\end{array}\right]$ and then $J$ is called a Jordan matrix with respect to . The set of such matrices is always a real vector space.
 • The command JordanMatrix(n, alg) returns a list of matrices which form a basis for the real vector space of  square matrices with entries in . With the keyword argument signature = [p, q] a basis for the Jordan matrices with respect to ${I}_{\mathrm{pq}}$ is determined.
 • The Jordan product of 2 Jordan matrices and is the symmetric product . The set of Jordan matrices with Jordan product is an algebra which is denoted by $\mathrm{𝕁}\left(n\mathit{,}\mathrm{𝔸}\right)$or .
 • The structure equations for a general Jordan algebra can be calculated with the command AlgebraData. The structure equations for a few low dimensional Jordan algebras are also available through the command AlgebraLibraryData.

Examples

 > with(DifferentialGeometry): with(LieAlgebras): with(Tools):

Example 1.

In this example we construct a basis for the Jordan algebra of matrices over the quaternions. The first step is to use the command AlgebraLibraryData to retrieve the structure equations for the quaternions.

 ${\mathrm{AD}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (2.1)

Initialize this algebra.

 > DGsetup(AD, '[e, i, j, k]', '[omega]');
 ${\mathrm{algebra name: Q}}$ (2.2)

Generate a basis for the Jordan algebra of matrices over the quaternions.

 > M := JordanMatrices(3, Q);
 ${M}{:=}\left[\left[\begin{array}{ccc}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {e}& {0}{}{e}\\ {e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\\ {e}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {e}\\ {0}{}{e}& {e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {i}& {0}{}{e}\\ {-}{i}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {j}& {0}{}{e}\\ {-}{j}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {k}& {0}{}{e}\\ {-}{k}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {i}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\\ {-}{i}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {j}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\\ {-}{j}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {k}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\\ {-}{k}& {0}{}{e}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {i}\\ {0}{}{e}& {-}{i}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {j}\\ {0}{}{e}& {-}{j}& {0}{}{e}\end{array}\right]{,}\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {k}\\ {0}{}{e}& {-}{k}& {0}{}{e}\end{array}\right]\right]$ (2.3)

We form the general element of $\mathrm{𝕁}\left(\mathit{3}\mathit{,}\mathrm{ℚ}\right)$ and check it is Hermitian.

 Q > C := [seq(c||n, n = 1 .. 15)];
 ${C}{:=}\left[{\mathrm{c1}}{,}{\mathrm{c2}}{,}{\mathrm{c3}}{,}{\mathrm{c4}}{,}{\mathrm{c5}}{,}{\mathrm{c6}}{,}{\mathrm{c7}}{,}{\mathrm{c8}}{,}{\mathrm{c9}}{,}{\mathrm{c10}}{,}{\mathrm{c11}}{,}{\mathrm{c12}}{,}{\mathrm{c13}}{,}{\mathrm{c14}}{,}{\mathrm{c15}}\right]$ (2.4)
 Q > J := evalDG(DGzip(C, M, "plus"));
 ${J}{:=}\left[\begin{array}{ccc}{\mathrm{c1}}{}{e}& {\mathrm{c4}}{}{e}{+}{\mathrm{c7}}{}{i}{+}{\mathrm{c8}}{}{j}{+}{\mathrm{c9}}{}{k}& {\mathrm{c5}}{}{e}{+}{\mathrm{c10}}{}{i}{+}{\mathrm{c11}}{}{j}{+}{\mathrm{c12}}{}{k}\\ {\mathrm{c4}}{}{e}{-}{\mathrm{c7}}{}{i}{-}{\mathrm{c8}}{}{j}{-}{\mathrm{c9}}{}{k}& {\mathrm{c2}}{}{e}& {\mathrm{c6}}{}{e}{+}{\mathrm{c13}}{}{i}{+}{\mathrm{c14}}{}{j}{+}{\mathrm{c15}}{}{k}\\ {\mathrm{c5}}{}{e}{-}{\mathrm{c10}}{}{i}{-}{\mathrm{c11}}{}{j}{-}{\mathrm{c12}}{}{k}& {\mathrm{c6}}{}{e}{-}{\mathrm{c13}}{}{i}{-}{\mathrm{c14}}{}{j}{-}{\mathrm{c15}}{}{k}& {\mathrm{c3}}{}{e}\end{array}\right]$ (2.5)

Here is the conjugate transpose of J.

 Q > Jdagger := DGconjugate(J)^+;
 ${\mathrm{Jdagger}}{:=}\left[\begin{array}{ccc}{\mathrm{c1}}{}{e}& {\mathrm{c4}}{}{e}{+}{\mathrm{c7}}{}{i}{+}{\mathrm{c8}}{}{j}{+}{\mathrm{c9}}{}{k}& {\mathrm{c5}}{}{e}{+}{\mathrm{c10}}{}{i}{+}{\mathrm{c11}}{}{j}{+}{\mathrm{c12}}{}{k}\\ {\mathrm{c4}}{}{e}{-}{\mathrm{c7}}{}{i}{-}{\mathrm{c8}}{}{j}{-}{\mathrm{c9}}{}{k}& {\mathrm{c2}}{}{e}& {\mathrm{c6}}{}{e}{+}{\mathrm{c13}}{}{i}{+}{\mathrm{c14}}{}{j}{+}{\mathrm{c15}}{}{k}\\ {\mathrm{c5}}{}{e}{-}{\mathrm{c10}}{}{i}{-}{\mathrm{c11}}{}{j}{-}{\mathrm{c12}}{}{k}& {\mathrm{c6}}{}{e}{-}{\mathrm{c13}}{}{i}{-}{\mathrm{c14}}{}{j}{-}{\mathrm{c15}}{}{k}& {\mathrm{c3}}{}{e}\end{array}\right]$ (2.6)

We see that J is Hermitian.

 Q > J &MatrixMinus Jdagger;
 $\left[\begin{array}{ccc}{0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\\ {0}{}{e}& {0}{}{e}& {0}{}{e}\end{array}\right]$ (2.7)

Now define two elements of $\mathrm{𝕁}\left(\mathit{3}\mathit{,}\mathrm{ℚ}\right)$ and calculate their Jordan product.

 Q > A := evalDG(M[8] + M[12]);
 ${A}{:=}\left[\begin{array}{ccc}{0}{}{e}& {j}& {k}\\ {-}{j}& {0}{}{e}& {0}{}{e}\\ {-}{k}& {0}{}{e}& {0}{}{e}\end{array}\right]$ (2.8)
 Q > B := evalDG(M[7] + M[15]);
 ${B}{:=}\left[\begin{array}{ccc}{0}{}{e}& {i}& {0}{}{e}\\ {-}{i}& {0}{}{e}& {k}\\ {0}{}{e}& {-}{k}& {0}{}{e}\end{array}\right]$ (2.9)
 Q > JordanProduct(A, B);
 $\left[\begin{array}{ccc}{0}{}{e}& \frac{{1}}{{2}}{}{e}& \frac{{1}}{{2}}{}{i}\\ \frac{{1}}{{2}}{}{e}& {0}{}{e}& \frac{{1}}{{2}}{}{j}\\ {-}\frac{{1}}{{2}}{}{i}& {-}\frac{{1}}{{2}}{}{j}& {0}{}{e}\end{array}\right]$ (2.10)

Example 2.

In this example we construct a basis for the Jordan matrices over the split octonions with respect to the inner product . First we retrieve the structure equations for the split octonions and initialize.

 Q > AD := AlgebraLibraryData("Octonions", Os, type = "Split");
 ${\mathrm{AD}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{e5}}{=}{\mathrm{e5}}{,}{\mathrm{e1}}{.}{\mathrm{e6}}{=}{\mathrm{e6}}{,}{\mathrm{e1}}{.}{\mathrm{e7}}{=}{\mathrm{e7}}{,}{\mathrm{e1}}{.}{\mathrm{e8}}{=}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e2}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e2}}{.}{\mathrm{e6}}{=}{\mathrm{e5}}{,}{\mathrm{e2}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e8}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e6}}{=}{\mathrm{e8}}{,}{\mathrm{e3}}{.}{\mathrm{e7}}{=}{\mathrm{e5}}{,}{\mathrm{e3}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e4}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e4}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{.}{\mathrm{e7}}{=}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e8}}{=}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e1}}{=}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e2}}{=}{\mathrm{e6}}{,}{\mathrm{e5}}{.}{\mathrm{e3}}{=}{\mathrm{e7}}{,}{\mathrm{e5}}{.}{\mathrm{e4}}{=}{\mathrm{e8}}{,}{{\mathrm{e5}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e5}}{.}{\mathrm{e6}}{=}{\mathrm{e2}}{,}{\mathrm{e5}}{.}{\mathrm{e7}}{=}{\mathrm{e3}}{,}{\mathrm{e5}}{.}{\mathrm{e8}}{=}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e1}}{=}{\mathrm{e6}}{,}{\mathrm{e6}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e6}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e6}}{.}{\mathrm{e4}}{=}{\mathrm{e7}}{,}{\mathrm{e6}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e6}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e6}}{.}{\mathrm{e7}}{=}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e1}}{=}{\mathrm{e7}}{,}{\mathrm{e7}}{.}{\mathrm{e2}}{=}{\mathrm{e8}}{,}{\mathrm{e7}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e7}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e7}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e7}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e7}}{.}{\mathrm{e8}}{=}{\mathrm{e2}}{,}{\mathrm{e8}}{.}{\mathrm{e1}}{=}{\mathrm{e8}}{,}{\mathrm{e8}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e8}}{.}{\mathrm{e3}}{=}{\mathrm{e6}}{,}{\mathrm{e8}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e8}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e8}}{.}{\mathrm{e6}}{=}{\mathrm{e3}}{,}{\mathrm{e8}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e8}}}^{{2}}{=}{\mathrm{e1}}\right]$ (2.11)
 ${\mathrm{algebra name: Os}}$ (2.12)

Here are the Jordan matrices we seek.

 Os > M := JordanMatrices(2, Os, signature = [1,1]);
 ${M}{:=}\left[\left[\begin{array}{cc}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\\ {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e1}}\\ {-}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e2}}\\ {\mathrm{e2}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e3}}\\ {\mathrm{e3}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e4}}\\ {\mathrm{e4}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e5}}\\ {\mathrm{e5}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e6}}\\ {\mathrm{e6}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e7}}\\ {\mathrm{e7}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e8}}\\ {\mathrm{e8}}& {0}{}{\mathrm{e1}}\end{array}\right]{,}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\\ {0}{}{\mathrm{e1}}& {\mathrm{e1}}\end{array}\right]\right]$ (2.13)

We form the general element of and check that it is Hermitian.

 Q > C := [seq(c||n, n = 1 .. 10)];
 ${C}{:=}\left[{\mathrm{c1}}{,}{\mathrm{c2}}{,}{\mathrm{c3}}{,}{\mathrm{c4}}{,}{\mathrm{c5}}{,}{\mathrm{c6}}{,}{\mathrm{c7}}{,}{\mathrm{c8}}{,}{\mathrm{c9}}{,}{\mathrm{c10}}\right]$ (2.14)
 Q > J := evalDG(DGzip(C, M, "plus"));
 ${J}{:=}\left[\begin{array}{cc}{\mathrm{c1}}{}{\mathrm{e1}}& {\mathrm{c2}}{}{\mathrm{e1}}{+}{\mathrm{c3}}{}{\mathrm{e2}}{+}{\mathrm{c4}}{}{\mathrm{e3}}{+}{\mathrm{c5}}{}{\mathrm{e4}}{+}{\mathrm{c6}}{}{\mathrm{e5}}{+}{\mathrm{c7}}{}{\mathrm{e6}}{+}{\mathrm{c8}}{}{\mathrm{e7}}{+}{\mathrm{c9}}{}{\mathrm{e8}}\\ {-}{\mathrm{c2}}{}{\mathrm{e1}}{+}{\mathrm{c3}}{}{\mathrm{e2}}{+}{\mathrm{c4}}{}{\mathrm{e3}}{+}{\mathrm{c5}}{}{\mathrm{e4}}{+}{\mathrm{c6}}{}{\mathrm{e5}}{+}{\mathrm{c7}}{}{\mathrm{e6}}{+}{\mathrm{c8}}{}{\mathrm{e7}}{+}{\mathrm{c9}}{}{\mathrm{e8}}& {\mathrm{c10}}{}{\mathrm{e1}}\end{array}\right]$ (2.15)

Here is the conjugate transpose of J.

 Q > Jdagger := DGconjugate(J)^+;
 ${\mathrm{Jdagger}}{:=}\left[\begin{array}{cc}{\mathrm{c1}}{}{\mathrm{e1}}& {-}{\mathrm{c2}}{}{\mathrm{e1}}{-}{\mathrm{c3}}{}{\mathrm{e2}}{-}{\mathrm{c4}}{}{\mathrm{e3}}{-}{\mathrm{c5}}{}{\mathrm{e4}}{-}{\mathrm{c6}}{}{\mathrm{e5}}{-}{\mathrm{c7}}{}{\mathrm{e6}}{-}{\mathrm{c8}}{}{\mathrm{e7}}{-}{\mathrm{c9}}{}{\mathrm{e8}}\\ {\mathrm{c2}}{}{\mathrm{e1}}{-}{\mathrm{c3}}{}{\mathrm{e2}}{-}{\mathrm{c4}}{}{\mathrm{e3}}{-}{\mathrm{c5}}{}{\mathrm{e4}}{-}{\mathrm{c6}}{}{\mathrm{e5}}{-}{\mathrm{c7}}{}{\mathrm{e6}}{-}{\mathrm{c8}}{}{\mathrm{e7}}{-}{\mathrm{c9}}{}{\mathrm{e8}}& {\mathrm{c10}}{}{\mathrm{e1}}\end{array}\right]$ (2.16)
 Os > I22 := Matrix([[1, 0], [0, -1]]);
 ${\mathrm{I22}}{:=}\left[\begin{array}{rr}{1}& {0}\\ {0}& {-}{1}\end{array}\right]$ (2.17)
 Os > evalDG(I22.J) &MatrixMinus evalDG(Jdagger.I22);
 $\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\\ {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\end{array}\right]$ (2.18)

Now define two elements of $\mathrm{𝕁}\left(\mathit{2}\mathit{,}\mathrm{ℚs}\right)$ and calculate their Jordan product.

 Q > A := evalDG(M[8] + M[10]);
 ${A}{:=}\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& {\mathrm{e7}}\\ {\mathrm{e7}}& {\mathrm{e1}}\end{array}\right]$ (2.19)
 Q > B := evalDG(M[1] + M[4]);
 ${B}{:=}\left[\begin{array}{cc}{\mathrm{e1}}& {\mathrm{e3}}\\ {\mathrm{e3}}& {0}{}{\mathrm{e1}}\end{array}\right]$ (2.20)
 Q > JordanProduct(A, B);
 $\left[\begin{array}{cc}{0}{}{\mathrm{e1}}& \frac{{1}}{{2}}{}{\mathrm{e3}}{+}\frac{{1}}{{2}}{}{\mathrm{e7}}\\ \frac{{1}}{{2}}{}{\mathrm{e3}}{+}\frac{{1}}{{2}}{}{\mathrm{e7}}& {0}{}{\mathrm{e1}}\end{array}\right]$ (2.21)