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矩陣方程的自反和反自反矩陣解
矩陣方程 的自反和反自反矩陣解
摘要：如果 滿足條件：（1） ，（2） ，則稱 為廣義反射矩陣，廣義反射矩陣也是自伴的對(duì)合矩陣。設(shè) 和 都是廣義反射矩陣，如果 滿足 ，則稱 為關(guān)于矩陣對(duì) 的廣義（反）自反矩陣；如果 滿足 ，則 稱為關(guān)于矩陣 的廣義（反）自反矩陣。這篇論文介紹了矩陣方程 ，在系數(shù)矩陣 ， 為廣義（反）自反矩陣的條件下，（反）自反矩陣解存在的充分必要條件及表達(dá)形式。另外，研究了矩陣方程 的（反）自反矩陣解集 ，利用矩陣的分解，導(dǎo)出（反）自反矩陣問(wèn)題的最佳逼近解。
關(guān)鍵詞：自反矩陣;反自反矩陣;矩陣方程;Frobenius范數(shù);矩陣最佳逼近問(wèn)題
The reflexive and anti-reflexive solutions of the
matrix equation
Abstract :An complex matrix is said to be a generalized reflection matrix if and .An complex matrix ia said to be a reflexive (or anti-reflexive) matrix with respect to the generalized reflection matrixs , if . An complex matrix ia said to be a reflexive (or anti-reflexive) matrix with respect to the generalized reflection matrix , if .This paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti-reflexive with respect to a generalized reflection matrixs solutions of the matrix equation .In addition, incorresponding solution set of the equation.The explicit expression of the nearest matrix to a given matrix in the Frobenius noum have been provided.
Keywords:Reflexive matrix; Anti-reflexive matrix; Matrix equation; Frobenius norm; Matrix nearness problem.
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