Polar Decomposition And Decomposition Of H-normal Matrices With Indefinite Inner Product

In the complex field, we discuss all kinds of decompositions of the H-normal matrices in detail (Theorems and its three corollaries). We also give the corresponding algorithm (section 4).Theorems Let XE.Fn×n is an H - normal matrix, then there exists a nonsingular matrix P, such thatWhere Xi and Hi have the same size, and each Xi has one or two distincteigenvalues.We use the canonical form of indecomposable H-normal matrices which aregiven by paper [1] and paper [2] in v(H)=l and v(H)=2 to obtain the canonical form of general H-normal matrices in v(H)<2 (theorem 9 and theorem 10; .Theorem 9 Let X be an H-normal matrix and v(H)=l, then there exists a nonsingular matrix P, such thatwhere v(H1) = 2,(X1,H1) is one of the pairs (4) (9), X2 isadiagonalmatrix.Note:you can see (4)⁹ in the text.Theorem 10 Let X be an H-normal matrix and v(H) = 2, then there exists a nonsingular matrix P, such thatwhere v(HJ)=2, (X1,Hl) is one of the pairs (10)²⁷, X2 is adiagonal matrix, Orwhere v(H1) =v(H2) = 1, (X1,H1) and (X2,H2) are one of the pairs (4)⁹, X3 is a diagonal matrix.We also research the logarithmic and exponential properties H-normal matrices (section 8)and study a several peculiar properties of the singular indecomposable H-normal matrices(section9 and section 10). We use these properties to completely solve open problem 1 and open problem 4 inv(//) z 2, which are advanced by Brianlines inthepaper[6] in 2001:Problem 1: Has every H-Normal matrix an H-polar decomposition? Problem 4: If an H - Normal matrices X admit an H - Polar decomposition X=UA, do the factors U and A commute?Moreover we answer the uniqueness of the H-polar decomposition. We obtain that nonsingular H-normal matrices all have an H-polar decomposition; moreover the factors U and A are commutable. But the singular H-normal matrices Xin v(H) < 2 finally turn to discuss their indecomposable factors Xi in v(Hi) < 2to use their decomposable theorem and canonical form. At last we use the knowledge of section 9 and section 10 (theorem 24 and theorem 26) to obtain only the second case may have no H-polar decomposition in v(H) <. 2, the others all have an H-polar decomposition, but their factors U and A are not commutable.Theorem24 Let X∈F2m×2m is a singular H-indecomposable matrix and hastwo distinct eigenvalues λ = 0 and μ≠ 0,if the canonical form of (X,H) isadmits an H-polar decomposition.Theorem26 Let the right side of equation (32) be nilpotent. If theJn can be represented as the disjoint union of two-element blocks. The two blocks such that: (1) the two blocks have the same size, one is m corresponding positive eigenvalues of H , the other is m correspondingnegative eigenvalues of H as n=2m; (2) the two blocks have the size different by one, one is m corresponding positive eigenvalues of H, the other is m-1 corresponding negative eigenvalues of H as n=2m-l, then there exists H -selfadjoint matrix ,4, such that: A2 =Jn.