Suppose that I have a program that can solve the system of linear equation $Ay=x$ efficiently when $A$ is a Hermitian matrix. I have a system of equations which is $(I+bH)y=x$, where $I$ is the identity matrix and $b$ is a complex scalar and $H$ is the hermitian matrix. Obviously $(I+bH)$ is not hermitian and neither is $(I/b+H)$. Can I reformulate this problem in terms of the Hermitian problem: $Ay=x$ ?
Reformulating a problem of solving a system of linear equations
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linear-algebra
matrices
1 Answers
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Introduce a variable $z=(I+bH)^*x$. Where $(I+bH)^*$ denotes the conjugate transpose of $(I+bH)$. Then solve $(I+bH)^*(I+bH)y=z$ using your method for Hermitian matrices since $(I+bH)^*(I+bH)$ is Hermitian.
$y$ doesn't necessarily solve your original problem, but if it doesn't, then it is a least squares solution to your equation if $(I+bH)^*(I+bH)$ is not singular.
If $(I+bH)^*(I+bH)$ is singular then $y=(I+bH)^+(I+bH)^{+*}z$ is a least squares solution to your original equation, where $A^+$ denotes the Moore-Penrose pseudoinverse of A.
Assuming your algorithm is coded correctly though, it should handle both of these cases already.
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0you are right for some reason I thought that $x$ the unknown. Sorry. – 2011-12-13