توابع ماتریسی تعمیم یافته، دترمینان و پرمننت

در این مقاله با استفاده از ماتریس های جایگشتی یا ماتریس های متقارن شرایط لازم و کافی برای اینکه یک تابع ماتریسی تعمیم یافته دترمینان یا پرمننت باشد، ارائه می شود. ثابت می کنیم که یک تابع ماتریسی تعمیم یافته، دترمینان یا پرمننت است اگر و فقط اگر حافظ ضرب ماتریس های جایگشتی متقارن باشد. همچنین نشان می دهیم که یک تابع ماتریسی تعمیم یافته، دترمینان است اگر و فقط اگر حافظ ضرب ماتریس های متقارن باشد.

generalized matrix functions, determinant and permanent

introductionsince linear and multilinear algebra has many applications in different branches of sciences, the attention of many mathematicians has been attracted to it in recent decades. the determinant and the permanent are the most important functions in linear algebra and so a generalized matrix function, which is a generalization of the determinant and the permanent, becomes significant. generalized matrix functions connect some branches of mathematics such as theory of finite groups, representation theory of groups, graph theory and combinatorics, and linear and multilinear algebra.    let  be the symmetric group of degree , be a subgroup of , and be a function. the function given byis called the generalized matrix function associated to  and . note that if and , the principal character of , then is the permanent and  if and , the alternating character of , then is the determinant.results    in this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. we prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. also we show that a generalized matrix function is the determinant if and only if it preserves the product of symmetric matrices. to be precise, we show that: if and is a nonzero function, then the following are equivalent:1)  or ;2) ;3) ;where is the permutation matrix induced by and .also if and is a nonzero function, then  if and only if for all symmetric matrices .