or
is defined by:
denote the space of column vector with n real components.
denotes the space of Column vectors with n complex
Components.
Vectors will be denoted by lower case letters. e.g.
if , then
where the are real.
will denote the unit column vector,all components being zero except the , which is
1 so that
e will denote the Column vector all of whose Components
are 1.
A matrix is a rectangular array of numbers in which not
only the value of the number is important but also its position in
the array. The size of the matrix is described by the number of
its rows and Columns.A matrix of n rows and m columns is said to
be nxm. Matrices will be denoted by upper Case letters.
e.g.
s an mxn matrix.
I or will denote the unit nxn matrix:
If for , we shall write
diag
Transposition of matrices(including vectors) will be denoted by a
prime or dash so that A' will be an nxm matrix and a' a row
vector. We will use * to denote Conjugate transposition so that
e.g.
and
We shall occasionally write to indicate the row of
a matrix and similarly the column of a matrix.
The inner product (or scalar product) of two vectors will be
denoted by
The determination of a square matrix A will be denote by det A.
The trace of a square matrix A is tr A =
The
characteristic polynomial of a square matrix A is det
. The characteristic values (or the eigen values) of A are the
zero's of its characteristic polynomial and are usually denoted by
and it is often assumed that
. In the
latter case, is called the dominant characteristic
(eigen) value and
the spectral radius of A. A
characteristic (eigen) vector of A corresponding to a
characteristic (eigen) value is a non-zero solution of the
equation
. We also call
as an eigen
(characteristic) pair of A.
: 2. Norms of Vectors
: lec1
: lec1
root
平成18年1月24日