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1. Some Notations:

$\delta (i,j)$ or $\delta_{ij}$ is defined by:

$$\delta (i,j)=0 \quad ;$$   if $$\,\, i\neq j$$

$$\hspace{1.7cm} 1 \quad ;$$   if$$\,\, i= j$$

$R_n$denote the space of column vectors with n real components.
$C_n$denotes the space of column vectors with n complex components.
Vectors will be denoted by lower case letters. e.g. if $a \in R_n$, then

$$a= \left[ \begin{array}{c} a_1\\ .\\ .\\ a_n\\ \end{array} \right]$$

where the are real.
$e_i$ will denote the $i^{th}$ unit column vector, all components being zero except the $i^{th}$, which is 1 so that $[e_i]_j=\delta(i,j)$ .
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 matrix. Matrices will be denoted by upper case letters.

e.g

is an mxn matrix.
I or $I_n$ will denote the unit nxn matrix:

$$[I]_{ij}=\delta (i,j)$$

If $a_{ij}=0$ for $i\neq j$, we shall write

$$A=$$ diag $$\,\,[a_{11}, ... a_{nn}]$$

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

 

and 

We shall occasionally write $R_i$ to indicate the $i^{th}$ row of a matrix and similarly $C_j$ the $j^{th}$ column of a matrix.

The inner product (or scalar product) of two vectors will be denoted by

$$(x,y)=x'y=\sum x_iy_i$$

The determination of a square matrix A will be denoted by det(A).
The trace of a square matrix A is tr A = $\sum a_{ii}$ The characteristic polynomial of a square matrix A is det $(A-\lambda I)$. The characteristic values (or the eigen values) of A are the zero's of its characteristic polynomial and are usually denoted by $\lambda_1 , ... \lambda _n$ and it is often assumed that $\vert\lambda_1\vert \geq \vert\lambda_2\vert\geq ...\geq \vert\lambda_n\vert$. In the latter case, $\lambda_1$ is called the dominant characteristic (eigen) value and $\vert\lambda_1\vert$ 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 $A x=\lambda x$. We also call $(\lambda,x)$ as an eigen (characteristic) pair of A.

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