Welcome to Wasin So's Quaternionic Page
The discovery of quaternion
 | Hamilton (1843) discovered quaternions after trying to extend complex numbers
to higher dimension for 10 years
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The development of quaternionic polynomial
| Hamilton (18??) proved that if the coefficients mutually commute then
the standard polynomial equation
has at most n distinct solutions (also commute with coefficients)
| Niven (1941) proved that a monic standard polynomial always has a root.
| Brand (1942) proved De Moivre's formula and used it to find nth roots of
a quaternion.
| Eilenberg and Niven (1944) proved that a generalized polynomial
with unique highest degree term always has a root.
| Kuiper and Scheelbeek (1959) gave another proof of Hamilton's result on
existence of root when coefficients commute
| Gordon and Motzkin (1965) proved that a monic standard polynomial
of degree n greater than or equal to 1 has either infinite or at most
n distinct roots.
| Beck (1979) gave another proof of Gordon and Motzkin's result on number of
roots of a monic standard polynomial
| Bray and Whaples (1983) gave another proof of Gordon and Motzkin's result
on number of roots of a monic standard polynomial
| Zhang and Mu (1994) obtained some roots of a standard quadratic
polynomial by solving a real linear system.
| Heidrich and Jank (1996) showed indirectly that quadratic equation has one,
two or infinite solutions.
| Porter (1997) solved linear polynomial explicitly.
He also derived formula for finding second solution of a quadratic
equation provided one solution is known.
| Cho (1998) re-proved De Moivre's formula for finding nth roots of
a quaternion.
| Huang and So (2002) derived explicit formulas for the roots of
a standard quadratic polynomial.
| Au-Yeung (2000) derived alternative explicit formulas for the roots of
a standard quadratic polynomial.
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The development of quaternionic eigenvalue
| Lee (1949) doubted the existence of left eigenvalue in general
| Cohen (1977) raised the question whether left eigenvalue always exists
| Wood (1985) proved the existence of left eigenvalue by a topological argument,
and reduced finding left eigenvalues of a 2-by-2 matrix to
solving a quadtratic quaternionic polynomial
| So (1995) reduced finding left eigenvalues of a 3-by-3 matrix to
solving a cubic quaternionic polynomial
| Huang and So (2001) provided an algebraic method of finding all left eigenvalues
of a 2-by-2 matrix by solving a quadtratic quaternionic polynomial explicitly
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The development of quaternionic numerical range
| Kippenhahn (1951) initiated the study of quaternionic numerical
range. BUT he WRONGLY concluded that quaternionic numerical range is
convex.
| Taussky (1954) proved some basic properties of quaternionic
numerical range. In particular, all right eigenvalues are contained in
the numerical ranges.
| Jamison (1972) observed that quaternionic numerical range is not
convex in general. He also showed that quaternionic numerical range is
convex for Hermitian matrix.
| Au-Yeung (1984) found an implicit characterization of matrices with
convex numerical range. When specializing to normal matrices, he obtained
a necessary and sufficient condition in terms of eigenvalues.
| So, Thompson and Zhang (1994) proved that the intersection of quaternionic
numerical range with the upper half complex plane is always convex for
normal matrix.
| Au-Yeung (1995) gave another proof for the fact that
the intersection of quaternionic
numerical range with the upper half complex plane is always convex for
normal matrix.
| So and Thompson (1996) proved that the intersection of quaternionic
numerical range with the upper half complex plane is always convex for
any matrix.
| So (1998) found an explicit characterization of matrices with
convex quaternionic numerical range.
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Last modified January 23, 2001
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