From: Janne Pesonen Subject: Re: Question on Quaternions Date: Thu, 11 Feb 1999 14:27:29 +0200 Newsgroups: sci.math Keywords: Solving equations in quaternions tom.kuo@chase.com wrote: > > I think attempting quadratics in quaternions is a bit ambitious. What about > > linear equations? Of course the general linear equation over the > > quaternions will be aX + Xb + c = 0. > > > > > Yes, quadratics in quaternions is by no means simple. As a matter of fact, > nobody even knows how to approach this thing. But I think the liner ones are > managable. [deletia --djr] BTW, a bibliography search gave these two promising sounding titles: Niven, I. Equations in quaternions. Amer. Math. Monthly 48 (1941), 654-661. Niven, I. The roots of a quaternion. Amer. Math. Monthly 49 (1942), 386-388. which may have something on quadratic equations (I do not know, if this is the case, since I don't have them). ============================================================================== [Remark: even solving linear equations is rather more complicated than shown here. The most general linear equation would be Sum( a_n X b_n ) = c for fixed quaternions {a_n}, {b_n}, and c. Probably the most efficient way to solve these would be to view X as an unknown vector in R^4, expand out the individual summands a_n X b_n as linear transformations M_n(X), then add. These equations may not have solutions, nor need they be unique. For example, iX-Xi=1 has no solutions, and iX-Xi=0 is true for all X in the linear span of 1 and i. In the particular case shown first, uniqueness is rather easy to check: if X1, X2 are solutions of aX+Xb+c = 0, then their difference X3=X1-X2 satisfies aX=-Xb. Assuming X1, X2 are distinct, X3 is invertible, giving X3^(-1) a X3 = (-b). This is a contradiction for most pairs {a, b}. --djr] ============================================================================== 5,169e 09.0X Eilenberg, Samuel; Niven, Ivan The "fundamental theorem of algebra" for quaternions. Bull. Amer. Math. Soc. 50, (1944). 246--248. A polynomial in quaternions is a sum of a finite number of monomials of the form $a\sb 0xa\sb 1xa\sb 2\cdots xa\sb k$, where $x$ and $a\sb i$ are real quaternions. It is here proved that, if such a polynomial has but one term of the highest degree $n$, then it always has at least one root. For the special case where each term is of the form $ax\sp k$ this had been proved by Niven [Amer. Math. Monthly 48, 654--661 (1941); these Rev. 3, 264] by essentially algebraic methods. The proof here given for the general case is topological. It is shown that the polynomial maps into itself the four-dimensional sphere of all real quaternions, with the point infinity added; and that the degree of this map (in the sense of Brouwer) is $n$. This is accomplished by showing that the given polynomial and the polynomial $x\sp n$ give homotopic maps and that the map of the latter has degree $n$. Reviewed by H. W. Brinkmann _________________________________________________________________ 4,67e 09.0X Niven, Ivan The roots of a quaternion. Amer. Math. Monthly 49, (1942). 386--388. The author investigates the number of solutions of the quaternion equation $\xi\sp m=\alpha$; the existence of such solutions is already known for any such algebraic equation [Amer. Math. Monthly 48, 654--661 (1941); these Rev. 3, 264]. The result is that there are exactly $m$ distinct $m$th roots of $\alpha$ if $\alpha$ is a quaternion that is not a real number; if $\alpha$ is real there are infinitely many $m$th roots unless $m=2$ and $\alpha$ is positive, in which case there are just the two square roots $±\surd\alpha$. The proof, which is very simple, gives an explicit determination of the roots in question. Reviewed by H. W. Brinkmann _________________________________________________________________ 3,264b 09.1X Niven, Ivan Equations in quaternions. Amer. Math. Monthly 48, (1941). 654--661. The following theorem is proved: If $D$ is a non-commutative division algebra with centrum $C$ then every equation of the form $x\sp m+a\sb 1x\sp {m-1}+a\sb 2x\sp {m-2}+\cdots+a\sb m=0 (a\sb m\neq 0)$ with coefficients in $D$ has a solution in $D$ if and only if $C$ is a real-closed field and $D$ is the algebra of real quaternions over $C$. The sufficiency of this condition follows from a general result of Ore [Ann. of Math. (2) 34, 480--508 (1933)] and the necessity was proved by R. Baer. The number of roots of such an equation is also completely discussed, by reducing the problem to the study of a quadratic equation. The result is that there may be infinitely many roots or a finite number, but in the latter case there are at most $(2m-1)\sp 2$. Reviewed by H. W. Brinkmann (c) 2001, American Mathematical Society