Some results on pinching matrices

by Ko, Chiu-chan

(Uncorrected OCR) Abstract of thesis entitled

SOME RESULTS ON PINCHING MATRICES

submitted by Chiu-Chan Ko

for the Degree of Master of Philosophy at The University of Hong Kong

in August 2003

An n x n matrix is said to be doubly stochastic (d.s.) if it has nonnegative real entries and all row sums and column sums are 1. The set of all n x n d.s. matrices will be denoted by Dn. A matrix P E Dn is called a pinching matrix, if it can be written as

t P=Q I-t

I-t t

for some permutation matrix Q and 0 :::; t :::; 1. Let P n be the set of all n x n d.s. matrices which can be expressed as finite products of pinching matrices.

Pinching matrices come up in the study of majorization and generalized numerical ranges. In fact, pinching matrices have been employed as a useful tool to study generalized numerical ranges.

Some basic results of finite products of pinching matrices are surveyed. Some results on necessary conditions are then given on the zero-nonzero pattern of a d.s. matrix A for A to be a finite product of pinching matrices. These results generalize a previously known result. The case of 3 x 3 d.s. matrices with exactly one zero entry is also considered, and a necessary and sufficient condition is given for such matrices to be finite products of pinching matrices.

The set of all 3 x 3 orthostochastic matrices has been proved by other au-

111

1

thors to be star-shaped with respect to the unique centre - 1 1 1 . The 3

111 analogous question for P3 is answered in this thesis. In particular, it is proved

111

1

that P3 is also star-shaped with respect to - 1 1 1 . Furthermore, a region 3

111 1 - 2b

b

b

of band c is given such that the matrix

b b

I-b-c

c

IS m

c

I-b-c

Finally, some open problems for further research are discussed.