Matrix Analysis
In Linear Algebra and applications, matrix analysis is the study of matrices and their algebraic properties. The following topics are my interests:
functions of matrices such as matrix exponentiation and matrix logarithm
eigenvalues and singular values of matrices with matrix decomposition and majorization
matrix norms
positive semi-definite and positive definite matrices
Operator Means
Average or mean is a very important notion that represents the property of data points. Kubo and Ando in 1980s have introduced operator mean of positive operators to generalize the notion of mean for positive numbers. Since their pioneer work on two-variable operator means, many different construction schemes of multi-variable geometric means have been developed. My interests are multi-variable geometric means of positive definite Hermitian matrices and the extension to positive operators:
Centroid
Ando-Li-Mathias symmetrization
Least squares mean
Quantum Information
Quantum information is the information of the state of a quantum system. It refers to both the technical definition in terms of von Neumann entropy and the general computational term. It is an interdisciplinary field that involves quantum mechanics, computer science, information theory, and cryptography among other fields. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, as an eigenstate in one basis is not an eigenstate in the other basis. As both variables are not simultaneously well defined, a quantum state can never contain definitive information about both variables.
Quantum information, like classical information, can be processed using digital computers, transmitted from one location to another, manipulated with algorithms, and analyzed with computer science and mathematics. Just like the basic unit of classical information is the bit, quantum information deals with qubits. Quantum information can be measured using von Neumann entropy.