It is well known that a Hermitian symmetric complex matrix A, [itex]A^{\dagger}=A[/itex] can be taking into a tridiagonolized form: [itex]A=V^{\dagger}HV[/itex] where [itex]^{\dagger}[/itex] is Hermitian conjugate and [itex]H[/itex] is the tridiagonal Hessenberg matrix, and [itex]V^{\dagger}V=VV^{\dagger}=I[/itex]. This decomposition is realized using Schmidt orthonormalization, where H is nothing but the coefficients generated by the orthonormaliztion.(adsbygoogle = window.adsbygoogle || []).push({});

Now suppose A is only symmetric, i.e., [itex]A^T=A[/itex] where T denotes transpose without conjugation, and proceed formally through Schmidt orthonormalization, where the corresponding "inner product" [itex]<x,y>=\sqrt{\sum_ix_i*y_i}[/itex] without taking abs value, i.e., not a real inner product, but can nevertheless be defined. The result would be that A is tridiagonalized into the form [itex]A=VHV^T[/itex] (if V has full rank), where [itex]V^TV=VV^T=I[/itex], enabling solution to the problem [itex]Ax=b[/itex] through solution of [itex]Hy=V^Tb[/itex] followed by [itex]x=Vy[/itex], where the H system is much easier to solve. Now my question is: where can this method go wrong? If so what is the condition on A under which this method does work?

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# A pseudo-orthogonalization process

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