Diagonalizing a non-Hermitian Matrix

*To*: mathgroup at smc.vnet.net*Subject*: [mg58741] Diagonalizing a non-Hermitian Matrix*From*: "Ituran" <isturan at gmail.com>*Date*: Sun, 17 Jul 2005 03:03:56 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hi, I have the following problem. There is a given non-Hermitian matrix M. Let's take an example in 2x2 dimension M={{1,2+4*I},{2-5*I,1}}. To diagonalize it we need 2x2 unitary matrices U and V such that U.M.Vdag = Md =Diag(3.8304,6.0273). Here Vdag = Transpose[Conjugate[V]]. To find U and V, we can work with the absolute square of the above eq, i.e, V.(Mdag.M).Vdag = U.(M.Mdag).Udag = M_D^2 = Diag(14.672,36.328). Again Mdag = Transpose[Conjugate[M]] and similarly for Udag. To find V, I can write the equation for the ith row of V; V_ij(Mdag.M)_jk = (M_D^2)_ii V_ik (no sum over i), k=1,2 in this case. (Mdag.M)_11 V_i1 + (Mdag.M)_21 V_i2 = (M_D^2)_ii V_i1, k=1 (Mdag.M)_12 V_i1 + (Mdag.M)_22 V_i2 = (M_D^2)_ii V_i2, k=2 Thus, to find the elements of V in the ith row(for both i=1 and 2) {valV,vecV} = Eigensystem[Transpose[Mdag.M]] where V = vecV (No transpose!). similarly for U {valU,vecU}=Eigensystem[Transpose[M.Mdag]] Of course, both vecV[[]] and vecU[[]] need to be normalized. I found V = {{-0.21954 + 0.49397*I,0.84130},{0.34169 - 0.76879I, 0.54056}} U ={{-0.34169 + 0.76879I*I,0.54056},{0.21954 - 0.49397*I,0.84130}} and they perfectly satisfy V.(Mdag.M).Vdag = U.(M.Mdag).Udag = Diag(14.672,36.328). However, when I check U.M.Vdag, I am getting U.M.Vdag = Diag(-3.8250 + 0.2031*I,6.0240 - 0.2031*I) where the absolute value of the entries are equal to the corresponding eigenvalues. The minus sign in front of 3.8250 is not a problem since U and and V are not unique(they contains some arbitrary phases which be used to get Md with nonnegative entries). So, the problem here is the existence of imaginary parts and I have no idea why I am getting such an answer. For example, for a Hermitian M (take 2-4*I instead of 2-5*I in (2,1) element of M), there is no such problem. What am I missing? Any idea? I am using Mathematica 4.0 in Windows 2K. Thanks a lot in advance, I.Turan

**Follow-Ups**:**Re: Diagonalizing a non-Hermitian Matrix***From:*Pratik Desai <pdesai1@umbc.edu>