Positive Definite Matrices
Chapter One
Positive Matrices
We begin with a quick review of some of the basic properties of positive
matrices. This will serve as a warmup and orient the reader to the
line of thinking followed through the book.
1.1 CHARACTERIZATIONS
Let H be the n-dimensional Hilbert space [C.sup.n]. The inner product between two vectors x and
y is written as (x,y) or as x*y. We adopt the convention that the inner product is conjugate
linear in the first variable and linear in the second. We denote by L(H) the space of all linear operators
on H, and by [M.sub.n] (C) or simply [M.sub.n] the space of n×n matrices with complex entries.
Every element A of L(H) can be identified with its matrix
with respect to the standard basis {[e.sub.j]} of [C.sub.n] . We use the symbol A for this matrix
as well. We say A is positive semidefinite if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
and positive ... read full excerpt from Positive Definite Matrices ebook
