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Register a new userBasic (nxm) real Matrix operations by using complex numbers
العمليات الرياضية على المصفوفات الحقيقية (nxm) باستخدام الأعداد العقدية
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نصر الدين عيد( باحث )
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في هذا العمل تم تعميم العمليات الرياضية على المصفوفات الحقيقيـة (2x2) و (3x3) و (4x4) التـي كانت قد درست من قبل (1996, 1993,1990, N.Ide) و ذلك على المصفوفات من المرتبـة (nxm) بغيـة تبسيط تلك العمليات و برمجتها على الحاسوب، ثم استخدامها في التطبيقات العمليـة مـستخدمين لـذلك أقواساً تضم أعداداً عقدية كتبت بعلاقات رياضية بينها و بين عناصر المصفوفات الحقيقية (إيزومـورفيزم بين هاتين الكتابتين)، و تم إجراء العمليات الرياضية على أعداد تلك الأقواس (العقدية) بدلاً من إجرائهـا على المصفوفات الحقيقية. كما تم ضبط عدد العمليات الرياضية و خاصة عملية الضرب باعتبارها الأكثـر كلفة للحاسوب. أخيراً تمت معالجة بعض الأمثلة العددية لإيجاد مقلوب المصفوفة (باعتبار هـذه العمليـة هي الأكثر تعقيداً على المصفوفات) بهذه الطريقة العقدية، و تمت مقارنة النتيجة التي حصلنا عليهـا بعـد برمجتها على الحاسوب بلغة الفورتران مع النتيجة الكلاسيكية لمقلوب المصفوفات و وجد أن الخطأ بـين النتيجتين كان شبه معدوم (جزء من مليون و أحياناً أقل) و من ثم فإن هذه الطريقة يمكن الاعتماد عليهـا في الحسابات و خاصة في المسائل الفيزيائية حيث يلاحظ في كثير من الحالات تطابق الكثير من الأعـداد المركبة لهذه المصفوفات الحقيقية؛ مما يخفض عدد العمليات الرياضية.
this paper, we generalize the study of the mathematical operations over (2x2), (3x3) and (4x4) real matrices by using the complex numbers which was introduced by (Ide. 1990,1993,1996), for any real matrices (nxm). This method of arithmetic operations of matrices by using the complex numbers and its properties is simple, easy and fast to program on computer. The importance of this method appears in the applications of problems which use the arithmetic operations of matrices, especially in physics.
References used
Ide, N. 1996. “Computer study of the mathematical operations over 2x2, 3x3 and 4x4 real matrices by using the complex numbers.”, 21st international conference on computer science & applications, Cairo, Egypt
Thomas Richard McCalla. 1966. "Introduction to Numerical Methods and Fortran Programing", John Wiley & Sons, Inc. New York
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