Find the matrix representations corresponding to complex numbers and quaternions.

[Hongyi Zhao]

How can I find the matrix representations corresponding to complex numbers and quaternions with the help of SageMath, i.e., the ring isomorphism from the field of complex numbers and quaternions to the rings of corresponding matrices, respectively, as described here and here?

Regards,

HZ

[John H Palmieri]

Is this the sort of thing you're looking for?

def matrix_rep(z):
    """
    INPUT: complex number z = a + bi
    OUTPUT: the matrix
       [a -b]
       [b  a]
    """
    a = z.real_part()
    b = z.imag_part()
    return matrix(RR, [[a, -b], [b, a]])

[Ray Rogers]

You (hongy) might be interested in
Matrix Groups (Universitext) 2nd Edition
by M. L. Curtis (Author)
Which is a pretty good introduction; although the price is a little high. 

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[Hongyi Zhao]

On Saturday, July 2, 2022 at 1:38:59 AM UTC+8 John H Palmieri wrote:

Is this the sort of thing you're looking for?

def matrix_rep(z):
    """
    INPUT: complex number z = a + bi
    OUTPUT: the matrix
       [a -b]
       [b  a]
    """
    a = z.real_part()
    b = z.imag_part()
    return matrix(RR, [[a, -b], [b, a]])

I want to identify the ring isomorphism between them programmatically, instead of defining a function based on this ring isomorphism.

Best,

HZ

[Hongyi Zhao]

On Saturday, July 2, 2022 at 2:49:25 AM UTC+8 raymond....@gmail.com wrote:

You (hongy) might be interested in
Matrix Groups (Universitext) 2nd Edition
by M. L. Curtis (Author)
Which is a pretty good introduction; although the price is a little high. 

Thank you for letting know this book [1]. The first chapter of the book talks about the problem discussed here.

[1] https://link.springer.com/book/10.1007/978-1-4684-0093-9

Best,

HZ