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

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.

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

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.

Last modified January 23, 2001