كليدواژه :
ﺿﺮب داﺧﻠﯽ ﻧﺎ ﻣﻌﯿﻦ , ﻣﺎﺗﺮﯾﺲ J - ﻫﺎوس ﻫﻮﻟﺪر , ﻣﺎﺗﺮﯾﺲ J- ﻣﺘﻌﺎﻣﺪ , ﻣﺎﺗﺮﯾﺲﻫﺎي J- ﻣﺘﻘﺎرن , ﺗﺠﺰﯾﻪ RQ
چكيده فارسي :
ﻫﻤﺎﻧﻄﻮر ﮐﻪ ﻣﯽداﻧﯿﻢ ﺗﺤﻮﯾﻞ ﯾﮏ ﻣﺎﺗﺮﯾﺲ ﺑﻪ ﺷﮑﻞ ﻓﺸﺮده ﯾﮑﯽ از ﻣﻬﻤﺘﺮﯾﻦ ﻣﺒﺎﺣﺚ ﻣﻄﺎﻟﻌﻪ ﻣﺎﺗﺮﯾﺲﻫﺎ ﺑﻪ ﺣﺴﺎب ﻣﯽآﯾﺪ. از اﯾﻦ دﺳﺖ ﻓﺮمﻫﺎي ﻓﺸﺮده ﻣﯽﺗﻮان ﺑﻪ ﻓﺮم ﺑﺎﻻ ﻫﺴﻨﺒﺮگ و ﻓﺮمﻫﺎي ﺳﻪ ﻗﻄﺮي ﺑﺮاي ﻣﺎﺗﺮﯾﺲﻫﺎي ﻣﺘﻘﺎرن اﺷﺎره ﮐﺮد، در اﯾﻦ راﺳﺘﺎ ﺑﺮاي رﺳﯿﺪن ﺑﻪ اﯾﻦ ﻓﺮمﻫﺎي ﻓﺸﺮده ﯾﮑﯽ از ﺑﻬﺘﺮﯾﻦ اﺑﺰارﻫﺎ ﻣﺎﺗﺮﯾﺲ ﻫﺎوس ﻫﻮﻟﺪر اﺳﺖ. دﯾﺪه ﻣﯽﺷﻮد ﮐﻪ ﺑﻮﺳﯿﻠﻪ ﻣﺎﺗﺮﯾﺲﻫﺎي ﻫﺎوس ﻫﻮﻟﺪر ﻣﯽﺗﻮان ﯾﮏ ﻣﺎﺗﺮﯾﺲ ﻣﺘﻘﺎرن را ﺑﻪ ﺷﮑﻞ ﺳﻪ ﻗﻄﺮي ﻣﺘﻘﺎرن ﺗﺤﻮﯾﻞ ﮐﺮد. اﯾﻨﮏ ﺑﺎ اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲﻫﺎي J- ﻫﺎوس ﻫﻮﻟﺪر اﯾﻦ ﮐﺎر را ﺑﺮاي ﻣﺎﺗﺮﯾﺲﻫﺎي J- ﻣﺘﻘﺎرن اﻧﺠﺎم ﻣﯽدﻫﯿﻢ.
در اﯾﻦ ﻣﻘﺎﻟﻪ ﻫﺪف اﺻﻠﯽ ﺗﻮﺟﻪ ﺑﻪ ﻣﺎﺗﺮﯾﺲﻫﺎي J- ﻫﺎوس ﻫﻮﻟﺪر و ﮐﺎرﺑﺮدﻫﺎﯾﯽ از آن اﺳﺖ، از دﺳﺘﺎوردﻫﺎي اﯾﻦ ﮐﺎر ﺗﺠﺰﯾﻪ RQ ﺑﺮاي ﯾﮏ ﻣﺎﺗﺮﯾﺲ ﻣﺮﺑﻌﯽ اﺳﺖ، ﮐﻪ در آن Q ﯾﮏ ﻣﺎﺗﺮﯾﺲ J- ﻣﺘﻌﺎﻣﺪ و R ﯾﮏ ﻣﺎﺗﺮﯾﺲ ﺑﺎﻻ ﻣﺜﻠﺜﯽ اﺳﺖ. ﺳﭙﺲ ﺗﺤﻮﯾﻞ ﻣﺎﺗﺮﯾﺲﻫﺎ ﺑﻪ ﻓﺮم ﻫﺴﻨﺒﺮگ ﺑﺮرﺳﯽ ﻣﯽﺷﻮد و ﺑﻌﺪ از آن ﻧﺸﺎن داده ﻣﯽﺷﻮد ﮐﻪ ﭼﮕﻮﻧﻪ ﯾﮏ ﻣﺎﺗﺮﯾﺲ J-ﻣﺘﻘﺎرن را ﻣﯽﺗﻮان ﺑﻪ ﻓﺮم ﺳﻪ ﻗﻄﺮي ﺗﺤﻮﯾﻞ ﮐﺮد.
چكيده لاتين :
The main concept in this paper is the notion of the J-Householder matrix and its main applications. From these cases are the achievement to QR-decomposition, where Q is a J-Orthogonal matrix and R is an upper triangular matrix and reduction to the Hessenberg form and the tridiagonal form, for J-symmetric matrices.
The reduction problem to condensed forms of triangular, Hessenberg and tridiagonal is one of the important problem in the numerical linear algebra. It is the
structures of these condensed forms that are exploited in the solution of the reduced problem. For example, as we have seen in [2], [3],[7], [8], [6], [9] and [10], the
solution of the linear system Ax = b is usually obtained by first triangularizing the
matrix A and then solving an equivalent triangular system. In [8], for reduction
to a condensed form, the concept of J−unitary similarity is used, while in the rest
is used in the ordinary sense. In eigenvalue computations, the matrix A is transformed to a Hessenberg form befor applying the QR iterations. In [1], for reduction
to a condensed form, the concept of J−unitary similarity is used. These condensed
forms are Householder transformations and mybe J−Householder transformations.
