Let $A$ be an $N$ by $N$ Hermitian matrix with elements $a_{ij}$. What will be the bound on the elements $b_{ij}$ where $B=A^{-1}$? If $A$ is a diagonal matrix, solution is trivial. Also for tri-diagonal matrix, bounds exists.
Bounds on inverse elements of Hermitian matrices
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linear-algebra
matrices
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0For tridiagonal matrices, bounds on inverse elements exist. Want to know about Hermitian matrices. – 2011-03-29
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1if you are more specific, you can (possibly) get more help... – 2011-03-29
1 Answers
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If H is a Hermitian matrix whose eigenvalues all have absolute value $\ge r$, then the matrix elements of $H^{-1}$ are bounded by $1/r$.
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0Can you provide a reference? – 2011-03-29
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3Diagonalize $H = U^* D U$ where $U$ is unitary, $D$ is diagonal and * is Hermitian conjugate. Moreover, $D = R |D|$ where $R$ is diagonal with diagonal elements $\pm 1$ (and thus unitary). Then for any unit column vectors $u$ and $v$, $u^* H^{-1} v = (R U u)^* |D|^{-1} (U v)$ where $RUu$ and $Uv$ are unit vectors. Now use Cauchy-Schwarz for the inner product $(x,y) \to x^* |D|^{-1} y$, and $x^* |D|^{-1} x \le (1/r) x^* x$. – 2011-03-30
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0Thanks for your answer. I didn't expect such a succinct proof sketch. – 2011-03-30