I have a question regarding the real part of some matrix A, defined as $$ Re\{A\} = \frac{1}{2}\left(A + A^\dagger \right).$$ Where $A^\dagger$ denotes the Hermitian conjugate. One can also assume that the real part is positive semi-definite, i.e. $Re\{A\} \geq 0$ .
Suppose I were to apply a similarity transformation with $S > 0$, and $S$ Hermitian to A as $S^{-1}AS$, what can be said about the real part of this similar matrix? Are there any bounds known of the form $$ Re\{S A S^{-1}\} \leq c(S) Re\{A\} $$ for some constant $c(S)$ that can depend on the matrix S? I would guess the constant $c(S)$ is somehow related to the largest eigenvalue of $S$.
The real part of a matrix under similarity transformation
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linear-algebra
matrices
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0When is a matrix $A$ smaller than a matrix $B$? Do you mean a norm of $\text{Re}\{M\}$? – 2012-04-03
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0Hi draks, Sorry I should have been more clear. No I meant in terms of the partial order for matrices, i.e. matrix $A \geq B$ if the difference between $A - B \geq 0$ is positive semi-definite. So to state the problem differently what is the smallest constant $c(S)$ so that for all vectors $\psi$ we have that $(\psi ,(c(S)Re\{A\} - Re\{SAS^{-1}\})\psi) \geq 0$. – 2012-04-03
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0It doesn't really have to be the smallest, any "reasonable" upper bound to $c(S)$ would do. – 2012-04-03