r_to_n {jordan}

Sizes of Matrix-based Jordan algebras

Description

Given the number of rows in a (matrix-based) Jordan object, return the size of the underlying associative matrix algebra

Usage

r_to_n_rsm(r)
r_to_n_chm(r)
r_to_n_qhm(r)
r_to_n_albert(r=27)
n_to_r_rsm(n)
n_to_r_chm(n)
n_to_r_qhm(n)
n_to_r_albert(n=3)

Arguments

n

Integer, underlying associative algebra being matrices of size $n\times n$

r

Integer, number of rows of independent representation of a matrix-based jordan object

Details

These functions are here for consistency, and the albert ones for completeness.

For the record, they are:

• Real symmetric matrices, rsm, $r=n(n+1)/2$, $n=(\sqrt{1+4r}-1)/2$

• Complex Hermitian matrices, chm, $r=n^2$, $n=\sqrt{r}$

• Quaternion Hermitian matrices, qhm, $r=n(2n-1)$, $n=(1+\sqrt{1+8r})/4$

• Albert algebras, $r=27$, $n=3$

Value

Return non-negative integers

Note

I have not been entirely consistent in my use of these functions.

Author(s)

Robin K. S. Hankin

Examples

r_to_n_qhm(nrow(rqhm()))