On Wed, Oct 31, 2018 at 11:39 PM Dan Asimov <dasimov@earthlink.net> wrote:
Thanks for the paper suggestions, Victor!
It occurs to me that as long as the matrices are over a ring M (assumed to have a multiplicative identity) in which the following hold:
----- a) 2 has a multiplicative inverse ...*
—and—
b) every element of Ring has a square root in Ring ... -----
... then a quadratic equation of form
X^2 + b X + c
for b, c in Ring and the unknown X to be found within Ring as well ...
... has a solution (OK, two) given by the usual formula:
----- X = -b/2 ± √((b/2)^2 - c).
I think this doesn't work because of non-commutativity. You can't factor X^2 + 2DX + D^2 as (X + D)^2 ( a necessary step in deriving the quadratic formula), because the latter expands to X^2 + DX + XD + D^2, which is different. Which raised the question of why the polynomials with the coefficients before the variables are the interesting ones. Shouldn't you be looking at the more general quadratic X^2 + AX + XB + C = 0?
-----
I'm guessing the cubic and quartic formulae for C work, too, for rings, that have some additional restrictions regarding characteristic, at least. (Cubics should definitely *not* be characteristic = 3 or we may have trouble dividing by 27.)
But commutativity may be a problem for degrees 3 and 4.
—Dan ————— * Here 2 denotes 1 + 1, in case you were wondering.
I wrote: ----- ... Consider a collection M of matrices over a ring, like
M = M(n, Z) or
M = M(n, Q) or
M = M(n, R) or
M = M(n, C),
meaning the ring of n x n square matrices over the integers, the rationals, the reals, or the complexes.
For any one of these, call it M and consider polynomials of form
P(X) = X^k + A_(k-1) X^(k-1) + ... + A_1 X + A_0
where the A_j belong to the matrix ring M.
Then what is known
a) about the existence of solutions X in M to the equation
P(X) = 0
(where 0 denotes the 0 matrix in M) ???
b) about closed formulas for the roots of P(X) in M ???
(like the quadratic, cubic, and quartic formulae).
Of course, there can't be any general quintic formula* for matrices, since that would contradict Abel's theorem on its unsolvability over C. -----
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