Fumio Hiai - Introduction to Matrix Analysis and Applications
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Matrices can be studied in different ways. They are a linear algebraic structure and have a topological/analytical aspect (for example, the normed space of matrices) and they also carry an order structure that is induced by positive semidefinite matrices. The interplay of these closely related structures is an essential feature of matrix analysis.
This book explains these aspects of matrix analysis from a functional analysis point of view. After an introduction to matrices and functional analysis, it covers more advanced topics such as matrix monotone functions, matrix means, majorization and entropies. Several applications to quantum information are also included.
Introduction to Matrix Analysis and Applications is appropriate for an advanced graduate course on matrix analysis, particularly aimed at studying quantum information. It can also be used as a reference for researchers in quantum information, statistics, engineering and economics.
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, 
denotes the space of all 
complex matrices. A matrix 
is a mapping 
. It is represented as an array with 
rows and 
columns: 
is the intersection of the 
th row and the 
th column. If the matrix is denoted by 
, then this entry is denoted by 
. If 
, then we write 
instead of 
. A simple example is the identity matrix 
defined as 
, or 
is a complex vector space of dimension 
. The linear operations are defined as follows: 
is a complex number and 
.
let 
be the 
matrix such that the 
-entry is equal to one and all other entries are equal to zero. Then 
are called the matrix units and form a basis of 
: 
and 
, then the product 
of 
and 
is defined by 
and 
. Hence 
. So 
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