Yik-Hoi Au-Yeung

3×3 Orthostochastic matrices and the convexity of generalized numerical ranges

Abstract Let U 3 be the set of all 3 × 3 unitary matrices, and let A and B be two 3 × 3 complex norḿal matrices. In this note, the authors first give a necessary and sufficient condition for a 3 × 3 doubly stochastic matrix to be orthostochastic and then use this result to consider the structure of the sets W ( A ) = {Diag UAU ∗ : U ∈ U 3 } and W ( A,B ) = {Tr UAU ∗ B : U ∈ U 3 }, where ∗ denotes the transpose conjugate.

A conjecture of marcus on the generalized numerical range

Let A be an n × n complex normal matrix with eigenvalues . In the present paper the authors show that if are not collinear, then the set is an orthonormal sct in is not convex which answers a conjecture posed by Marcus concerning the convexity of the generalized numerical range.

Some theorems on the generalized numerical ranges

Let F be the real field R the complex field C or the skew field H of real quaternions. For any c = i:c1,…,cn )∊ Rn: and for any two hermitian (symmetric) matrices A and B with elements in F, the authors give a unified proof that the set is convex (with n> 2 if F =R). They also show that the convexity problem, definite-ness problem, and inclusion problem associated with the above set, are all equivalent, and give some applications of these results.

On the convexity of numerical range in quaternionic hilbert spaces

Let A be a bounded linear operator in a quaternionic Hilberi space (H,(·,·)). The numerical range of A is defined to be the set W (A)={( Au, u): u e H, (u,u) = 1}. Quite different from the complex caseW (A) may not be convex. In this note the author proves that W (A) is convex if and only if R ∩ W (A) = {Req : q e W(A)}. where R is the real field and Re q denotes the real part of the quaternion q. For a normal operator A in a finite dimensional space H, the author gives a characterization on the convexity of W(A) in terms of ihe eigenvalues of A and also proves that the generalized numerical range of A is convex if and only if A is Hermitian.

A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

A simple proof of the convexity of the field of values defined by two hermitian forms

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Permutation matrices whose convex combinations are orthostochastic

Abstract Let P 1 ,…, P m be n × n permutation matrices. In this note, we give a simple necessary condition for all convex combinations of P 1 ,…, P m to be orthostochastic. We show that for n ⩽15 the condition is also sufficient, but for n ⩾15 whether the condition is sufficient or not is still open.

On the semi-definiteness of the real pencil of two Hermitian matrices

Abstract Let A and B be two n × n real symmetric matrices. A theorem of Calabi and Greub-Milnor states that if n ⩾3 and A and B satisfy the condition (uAu′) 2 + (uBu′) 2 ≠ 0 for all nonzero vectors u , then there is a linear combination of A and B that is definite. In this note, the author proves two theorems of the semi-definiteness of a nontrivial linear combination of A and B by replacing the condition ( ∗ ) by another condition. One of these theorems is a generalization of the theorem of Greub-Milnor and Calabi.

Two formulas for the generalized radon-hurwitz number *

The maximal numberk, of m × n real matrices Ei satisfying is known as the generalized Radon-Hurwitz number, ρ(m,n). In this note, ρ(m,n) is evaluated for some particular values of m and n.