หลักทางคณิตศาสตร์ > Matrices

SQUARE MATRICES.  When m  =  n,  (1.1)  is square and will be called a square matrix of        order n or an  n – square matrix.

In a square matrix, the elements  a11 , a22 , .... , ann  are called its diagonal elements.

The sum of the diagonal elements of a square matrix  A  is called the trace of  A.

 

ZERO  MATRIX.         A  matrix, every element of which is zero, is called a zero matrix.   When                 A  is a zero matrix and there can be no confusion as to its order, we shall write  A  =  0          instead of the  m  x  n  array of zero elements.

 

SUMS  OF  MATRICES.        If  A  =  [aij ]  and  B  =  [bij ]  are  two  m  x  n  matrices,  their  sum                (difference),  A ± B ,  is  defined as the  m  x  n  matrix  C  =  [cij ] ,  where each element               of  C  is  the  sum  (difference)  of  the  corresponding elements of  A  and  B.  Thus,               A ± B  =  [aij ± bij ].

 

THE  IDENTITY  MATRIX. A square matrix  A  whose elements  aij  =  0  for  i >  j  is called       upper triangular ;  a square matrix  A  whose elements  aij  =  0  for  i < j  is called     lower triangular.  Thus

 

SPECIAL  SQUARE  MATRICES.  If  A  and  B  are square matrices such that AB  =  BA ,         then  A  and  B  are called commutative or are said to commute.  It is a simple matter to          show that if  A  is any  n-square matrix,  it commutes with itself and also with  In.

 

THE  INVERSE  OF  A  MATRIX.   If  A and B  are  square matrices such that  AB  =  BA  =  I,                  then  B  is called the inverse of  A  and we write  B  =  A-1  (B  equals  A  inverse).          The         matrix  B  also has  A  as its inverse and we may write  A  =  B-1 .

 

THE  TRANSPOSE  OF  A  MATRIX.       The matrix of order  n x m  obtained by interchanging            the rows and columns of an  m x n  matrix  A  is called the transpose of  A  and is       denoted by A¢  (A  transpose).

 

SYMMETRIC  MATRICES.       A square matrix  A  such that  A¢ =  A  is called symmetric.      Thus,  a square matrix  A  =  [aij ]  is symmetric provided  aij  =  aji ,  for all values of  i         and  j.

                A square matrix  A  such that  A¢ =  -A  is called skew-symmetric.

 

HERMITIAN  MATRICES.  A square matrix  A  =  [aij ]  such that  Ā¢ =  A  is called    Hermitian.   Thus,  A  is Hermitian provided  aij  =  äij  for all values of  i  and  j .         Clearly,  the diagonal elements of an Hermitian matrix are real numbers.

 

THE  RANK  OF  A  MATRIX.       A non-zero matrix  A  is said to have rank  r  if at least one of its  r-square minors is different from zero while every  (r + 1)-square minor,  if any,  is       zero.   A zero matrix is said to have rank 0.

 

EQUIVALENT  MATRICES.      Two matrices  A  and  B  are called equivalent,  A~B,   if one   can be obtained from the other by a sequence of elementary transformations.

 

ROW  EQUIVALENCE.        If a matrix  A  is reduced to  B  by the use of elementary row           transformations a lone,  B  is said to be row equivalent to  A  and conversely.   The   matrices  A  and  B  of Example  3  are row equivalent.

 

THE  NORMAL  FORM  OF  A  MATRIX.    By means of elementary transformations any      matrix  A  of rank  r > 0  can be reduced to one of the forms

 

ELEMENTARY  MATRICES.       The matrix which results when an elementary row  (column)  transformation id applied to the identity matrix  In  is called an elementary row  (column)  matrix.   Here,  an elementary matrix will be denoted by the symbol introduced to denote the elementary transformation which produces the matrix.






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