A2.1 Vectors and Matrices

An N K matrix A is an array of numbers as follows:

where a _n,k denotes the ( n, k)th element. If N = K, the matrix A is called square. An N 1 vector a is a matrix with one column:

where a _n denotes the nth element of a.

Basic manipulations with matrices are as follows.

1. Addition:

   
   

   where the sizes of A, B, and C are the same.

2. Multiplications:

   • For A and B of size N M and M K, respectively:

     
     

     where the size of C is N K.

   • For a scalar ?:

     
     

1. Transpositions:

   • The transpose of A, denoted by A ^T, is defined by

     
     

     where [ A] _n,k stands for the ( n, k)th element of A. We also have

     
     

   • The Hermitian transpose of A, denoted by A ^H, is defined by

     
     

     where the superscript * denotes the complex conjugate. We also have

     
     

1. Triangular matrices: A is called lower triangular if

   
   

   If the diagonal elements of A are all zeros, A is called strictly lower triangular.

   A is called upper triangular if

   
   

   If the diagonal elements of A are all zeros, A is called strictly upper triangular.

2. Symmetric: A square matrix A is called...