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【摘要】给出了复亚正定矩阵的两个行列式不等式，推广了[2]之相应结论。
【关键词】复亚正定矩阵；行列式；不等式
Several Determinant Inequalities of Complex Metapositive Definite Matrices
Yuan Nanqiao
(Sichuan University of Arts and Science， Dazhou Sichuan，635000，China)
【Abstract】In this paper we obtain two determinant inequalities of the complex metapositive definite matrices，these results generalize corresponding results in[2].
【Key words】complex metapositive definite matrix；determinant；inequalities
【CLC number】O151.21
【Document code】A
【Article ID】1005-250X(2007)11-0085-03
1 Introduction
In[1]，[2]，[3]and[7]，[8] discussed several theories of real general positive definite matrices，became an important production of the development of positive definite matrices theories. In [4] discussed generalizer of Hermitian positive definite matrix on complex number field，give out the complex metapositive definite matrix’s basic quality and equivalent express. This paper discussed several determinant inequalities of complex metapositive definite matrices，generalize corresponding results in [2].
In this paper，we let C n×n be all n-th-order square matrix. Let HDP(HDP0) be all n-th-order Hermitian positive definite (positive semi-definite) matrices. Let A* be adjoint matrix of A.
Definition 1.1 Let A=(a ij ∈C n×n，if each x∈Cn，x≠0，and all must be Re>0(x*Ax)>0(≥0)，at the situation is called A complex metapositive definite (complex metapositive semi-definite) matrices，marked A∈CPD(CPD0).
If we denoted the set of n-th-order real metapositive definite matrixes by RPD and n-th-order real symmetric positive definite matrices by SPD. Then
SPDRPDCPD
andSPDHPDCPD.
2 Main results
Theorem 2.1 Let A∈CPD，B∈HPD，then we have
|det(A+B)|>|det A|+|det B|
=|det A|+det B
Corollary 2.1.1 Let A∈CPD，B∈HPD， then we have
|det(A+B)|1n>|det A|1n+|det B|1n
Corollary 2.1.2 Let A∈CPD0，B∈HPD，then we have
|det(A+B)|≥|det A|+|det B|
Theorem 2.2 Let A∈CPD，B∈HPD0，and 0 |det(A+B)|≥|det A|
Corollary 2.2.1 Let A∈CPD0，B∈HPD0，and0 |det(A+B)|≥|det A|
3 Some Lemmas[4]
Lemma 3.1 Let A∈Cn×n，Q∈Cn×n and Q be nonsingular matrix，then we have
Q*AQ∈CPDA∈CPD
Lemma 3.2 Let A∈CPD，then A be positively stable matrix.That is real part of all proper value of A is positive number.
Lemma 3.3 Let A=AkA1
4 The proofs of Theorem 2.1 and Theorem 2.2
Proof Theorem 2.1 since B∈HPD we can see，there exist nonsingular matrix P∈Cn×n，such that B=P*P[5]. Obviously
We can also find out from lemma 3.1 Q*AQ∈CPD. And by Lemma 3.2 we can see there exist n-th-order unitary matrix U∈Un×n，such that
U*(Q*AQ)U=λ1*
Since det B>0，so det(Q*Q)>0.We divide each side of the inequality (1) by det(Q*Q)，Theorem 2.1 is proved.
For example，asserts that if A=H+iG and B are given such that H and B are positive definite，and G is Hermitian，then we may replace (A，B ) by (B1/2BA-1/2，In) and assume that B=In，Since A+A* is positive definite，the eigenvalues a1，…，an of A have positive real parts. Thus，
Similarly Theorem 2.1 we can also prove Corollary 2.1.1 and 2.1.2. Pay attention to when A∈RPD，Theorem 2.1 is the results in [2]，when A∈HPD，is classical conclusion. By all appearance，these corollary generalize corresponding results in [2].
Proof Theorem 2.2 Since B∈HPD0 and 0
References：
[1] TU Bo-xun. Theory of general positive definite matrices (Ⅰ) [J]. Acta. Math. Sinica，1990，33(4)：462~471.(in Chinese)
[2] TU Bo-xun. Theory of general positive definite matrices (Ⅱ) [J]. Acta. Math. Sinica，1991，34(1)：91~102.(in Chinese)
[3] YANG Xin-min. One determinant inequation of general positive definite matrices [J]. Math. in Practice and Theory，1994，24(2)：45~47.(in Chinese)
[4] FENG Ming-xian. Complex metapositive definite matrices [J]. Northeast Math，1994(1)：51~53. (in Chinese)
[5] YANG Ke-xun，BAO Xue-you. Matrix analyse [M]. Ha’erbing： Ha’erbing Industry University Press，1995.96.(in Chinese)
[6] Marshall A. W. and Olkin I.，Inequalities： Theory of Majoriazation and its Applications [M].Academic Press，(1979).
[7] Pullman N. J.，Matrix Theory and Applications [M]. Marcel Dekker Inc，New York and Basel.，1976，209~227.
[8] HE Gan-tong，Several determinant inequalities of positive definite Hermittian matrices [J]. J.of Math. Res. & Expo.，2002，22(1)：79~82(in Chinese)
Received date：2007-10-11
注：“本文中所涉及到的图表、注解、公式等内容请以PDF格式阅读原文。”
【关键词】复亚正定矩阵；行列式；不等式
Several Determinant Inequalities of Complex Metapositive Definite Matrices
Yuan Nanqiao
(Sichuan University of Arts and Science， Dazhou Sichuan，635000，China)
【Abstract】In this paper we obtain two determinant inequalities of the complex metapositive definite matrices，these results generalize corresponding results in[2].
【Key words】complex metapositive definite matrix；determinant；inequalities
【CLC number】O151.21
【Document code】A
【Article ID】1005-250X(2007)11-0085-03
1 Introduction
In[1]，[2]，[3]and[7]，[8] discussed several theories of real general positive definite matrices，became an important production of the development of positive definite matrices theories. In [4] discussed generalizer of Hermitian positive definite matrix on complex number field，give out the complex metapositive definite matrix’s basic quality and equivalent express. This paper discussed several determinant inequalities of complex metapositive definite matrices，generalize corresponding results in [2].
In this paper，we let C n×n be all n-th-order square matrix. Let HDP(HDP0) be all n-th-order Hermitian positive definite (positive semi-definite) matrices. Let A* be adjoint matrix of A.
Definition 1.1 Let A=(a ij ∈C n×n，if each x∈Cn，x≠0，and all must be Re>0(x*Ax)>0(≥0)，at the situation is called A complex metapositive definite (complex metapositive semi-definite) matrices，marked A∈CPD(CPD0).
If we denoted the set of n-th-order real metapositive definite matrixes by RPD and n-th-order real symmetric positive definite matrices by SPD. Then
SPDRPDCPD
andSPDHPDCPD.
2 Main results
Theorem 2.1 Let A∈CPD，B∈HPD，then we have
|det(A+B)|>|det A|+|det B|
=|det A|+det B
Corollary 2.1.1 Let A∈CPD，B∈HPD， then we have
|det(A+B)|1n>|det A|1n+|det B|1n
Corollary 2.1.2 Let A∈CPD0，B∈HPD，then we have
|det(A+B)|≥|det A|+|det B|
Theorem 2.2 Let A∈CPD，B∈HPD0，and 0
Corollary 2.2.1 Let A∈CPD0，B∈HPD0，and0
3 Some Lemmas[4]
Lemma 3.1 Let A∈Cn×n，Q∈Cn×n and Q be nonsingular matrix，then we have
Q*AQ∈CPDA∈CPD
Lemma 3.2 Let A∈CPD，then A be positively stable matrix.That is real part of all proper value of A is positive number.
Lemma 3.3 Let A=AkA1
4 The proofs of Theorem 2.1 and Theorem 2.2
Proof Theorem 2.1 since B∈HPD we can see，there exist nonsingular matrix P∈Cn×n，such that B=P*P[5]. Obviously
We can also find out from lemma 3.1 Q*AQ∈CPD. And by Lemma 3.2 we can see there exist n-th-order unitary matrix U∈Un×n，such that
U*(Q*AQ)U=λ1*
Since det B>0，so det(Q*Q)>0.We divide each side of the inequality (1) by det(Q*Q)，Theorem 2.1 is proved.
For example，asserts that if A=H+iG and B are given such that H and B are positive definite，and G is Hermitian，then we may replace (A，B ) by (B1/2BA-1/2，In) and assume that B=In，Since A+A* is positive definite，the eigenvalues a1，…，an of A have positive real parts. Thus，
Similarly Theorem 2.1 we can also prove Corollary 2.1.1 and 2.1.2. Pay attention to when A∈RPD，Theorem 2.1 is the results in [2]，when A∈HPD，is classical conclusion. By all appearance，these corollary generalize corresponding results in [2].
Proof Theorem 2.2 Since B∈HPD0 and 0
References：
[1] TU Bo-xun. Theory of general positive definite matrices (Ⅰ) [J]. Acta. Math. Sinica，1990，33(4)：462~471.(in Chinese)
[2] TU Bo-xun. Theory of general positive definite matrices (Ⅱ) [J]. Acta. Math. Sinica，1991，34(1)：91~102.(in Chinese)
[3] YANG Xin-min. One determinant inequation of general positive definite matrices [J]. Math. in Practice and Theory，1994，24(2)：45~47.(in Chinese)
[4] FENG Ming-xian. Complex metapositive definite matrices [J]. Northeast Math，1994(1)：51~53. (in Chinese)
[5] YANG Ke-xun，BAO Xue-you. Matrix analyse [M]. Ha’erbing： Ha’erbing Industry University Press，1995.96.(in Chinese)
[6] Marshall A. W. and Olkin I.，Inequalities： Theory of Majoriazation and its Applications [M].Academic Press，(1979).
[7] Pullman N. J.，Matrix Theory and Applications [M]. Marcel Dekker Inc，New York and Basel.，1976，209~227.
[8] HE Gan-tong，Several determinant inequalities of positive definite Hermittian matrices [J]. J.of Math. Res. & Expo.，2002，22(1)：79~82(in Chinese)
Received date：2007-10-11
注：“本文中所涉及到的图表、注解、公式等内容请以PDF格式阅读原文。”