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()))