ALGORITHMS FOR OPERATIONS ON PROJECTION-MATRICES OF ( ) CnM

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ALGORITHMS FOR OPERATIONS ON PROJECTION-MATRICES OF ( ) CnM

Journal Title: Наукові праці. Серія "Комп’ютерні технології" - Year 2016, Vol 287, Issue 275

Abstract

The rapid development of integrated systems of computer mathematics for PCs affected most intellectual sphere of human activity – especially solving complex mathematical and scientific-technical problems, problems of field theory, aerodynamics, space, mathematical modeling systems and so on. But math methods are in constant evolutionary development and therefore the latest achievements of mathematical scientific schools are not covered by such integrated systems. One of the areas of mathematical tools – ∗C -operator algebra. This is due to the needs and demands of modern physics. In this paper, it’s the matrix of special type: m mM ⋅ = ∗ . Matrix M is projection if and only if 1 2 2 2 2 1 = +++= n mmmm  . For this matrix presented algorithm of conversion to within unitary transformation from a complexvalued matrix-projection to real-valued matrix-projection: 1) obtaining initial data; 2) verification of source data; 3) schedule in the form projection; 4) finding out: m mMU ˆ ˆˆ , ⋅ = ∗ ; 5) output results. And matrix properties of this type are investigated. That, matrix m mM ⋅ = ∗ , s sS ⋅ = ∗ , ( ) 0001 1  += kk mmmm , ( ) nkk ssss 1 000   += , 0 ≠⋅∗ sm are orthogonal 0== SMMS if and only if 0 11 =+ ++ kkkk smsm . And let matrix ( ) CnMDM ∈, with complex elements are as follows: m mM ⋅ = ∗ , d dD ⋅ = ∗ , where C N ∈ =∈∀ i i dm,ni , 1 , 1 2 = m , 1 2 =d then M MDM 2 τ = if and only if 2 2 τ =⋅ ∗dm . For family of ( ) ( ) ( ) kmkmkM ⋅= ∗ matrix, where ( ) ( ) ( ) n n kn kk k kmkmkm C ∈         = −− + −         1 1 1 0,,0,~,~,0,,0 presented algorithm of conversion to within unitary transformation from a complex-valued family of matrix-projection to real-valued family of matrix-projection and calculating ( ) MF ~ -function of family this matrix: 1) obtaining initial data; 2) verification of source data; 3) schedule in the form projection; 4) finding out: ( ) ( ) ( ) kmkmkMU ˆˆˆ , ⋅= ∗ ; 5) calculating ( ) MF ~ ; 6) output results. So, a algorithmization of basic operations of objects which are the matrix-projection and the family of the matrix-projections was made.

Authors and Affiliations

E. Samoilenko

Keywords

integrated systems; computer mathematics; operator; algebra; matrix; projection; algorithm

EP ID EP225099
DOI -
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