In the vector space $V$ of $3\times 3$ symmetric real matrices, we can define a nonassociative algebra structure by the multiplication $$A \bullet B = \frac12 (AB +BA).$$ This turns $V$ into a Jordan algebra.
What is the minimum number of generators of this Jordan algebra? And could you give me one set of such generators?
Thanks a lot!
The minimum number of generators is 2. First, it is easy to check that one generator is not enough: every symmetric matrix is diagonalizable, so the subalgebra it generates has dimension at most 3.
Next, the claim is that $$A:=S_{11}+2S_{22}+3S_{33}$$ and $$B:=S_{12}+2S_{13}$$ generate the algebra, where $S_{ij}$ is the usual notation for the symmetric matrices. In fact the powers of $A$ generate the subalgebra $D$ of diagonal matrices (by Vandermonde) and the powers $B$ satisfy $B^2=S_{23}$ mod $D$ and $B^3=S_{12}+\frac12S_{13}$ mod $D$.
Although Andrei Moroianu has already answered the question, I would like to add an answer I just received from Efim Zelmanov. The Jordan algebra is generated by two matrices. The first is the matrix unit $E_{1,1}$, and the second any "generic" symmetric $3\times 3$-matrix. I checked: it works if the second matrix $$\left\[\begin{array}{ccc} 1 & 2 &3 \\\ 2 & 4 & 5 \\\ 3 & 5 & 6\end{array}\right\].$$
Update 1. An interesting question is whether it is true for $4\times 4$-symmetric matrices. The answer seems to be "yes". How about arbitrary dimension $n\ge 3$?
Update 2. This works for any field.
This non-associative algebra is the typical example of a Jordan algebra, named after Pascual Jordan, who studied them with J. von Neumann. You should consult a dedicated book. See for instance J. Faraut & A. Korányi: Analysis on symmetric cones, Oxford University Press, New York, 1994.
A complete classification of the so-called Euclidian case is available, but it was disappointing in view of von Neumann & Jordan's expectation.
This is already well answered, many choices of two symmetric matrices will do. I'll add this observation: $\{I,A,B,A \bullet A,B \bullet B,A\bullet B\}$ is an additive basis where $$A=\left[\begin{array}{ccc} 1 & 1 &1 \\ 1 & 0 & 0 \\ 1 & 0 & 0\end{array}\right] \qquad B=\left[\begin{array}{ccc} 0 & 0 &1 \\ 0 & 0 & 1 \\ 1 & 1 & 1\end{array}\right]$$ Note that $A\bullet B$ has rank $3$, not that it matters. I'm not sure if two rank 1 matrices could generate the algebra, but I lean against (I couldn't get past dimension 4), I didn't look hard for a generating set with a rank 1 and a rank 2 matrix.
