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I am searching for any estimates of the greatest eigenvalue for non-symmetric 3(5)-diagonal matrix $A$, i.e. any information about estimates like

$$|\lambda_n|<F(a_{ij}), $$ where $A=[a_{ij}];i,j=\overline{1,n},F$- any function. This estimate will be used in computations, so the better it is, the quicker the computation is. Could you advise any source? Any help will be sincerely appreciated

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  • $\begingroup$ What do you mean by "$3(5)$-diagonal"? $\endgroup$ Jan 5, 2015 at 23:03
  • $\begingroup$ I mean tridiagonal matrix, something like $$\left[ \begin{matrix} {a_{11}} & a_{12} &{0} & {0}& {..} &{..} \\ {a_{21}} & {a_{22}} & {a_{23}}& {0} &{..} \\ {0}&{a_{32}}&{a_{33}}&{a_{34}}&{0} &{0} &{..}\\ {..} \end{matrix} \right]$$ 5-diagonal means almost the same, but there are 5 elements in each line (except some at the beginning and the end) $\endgroup$
    – cool
    Jan 5, 2015 at 23:08

2 Answers 2

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Since the spectral radius $\rho(A)$ can be bounded from above by any matrix norm consistent with a certain vector norm (e.g., a matrix operator norm), we have $$ \rho(A)\leq\min\{\|A\|_1,\|A\|_{\infty}\}. $$ This bound can be also obtained by using the Geršgorin theorem on $A$ and $A^T$. Since the $1$-norm and the $\infty$-norm involve maxima of absolute column/row sums (and, in addition, they are easy to compute), they are likely to provide a reasonably accurate estimate for sparse/banded matrices.

In addition, as far as I know, the bound is asymptotically exact for symmetric 3-diagonal Toeplitz matrices (with $n\rightarrow\infty$).

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  • $\begingroup$ Indeed, this estimate is better, than with Frobenius norm (I checked this :) ) Thanks, you greatly helped! $\endgroup$
    – cool
    Jan 7, 2015 at 0:49
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$|\lambda_n| \le \|A\|$ for any induced matrix norm. There are many estimates of these, e.g. you could bound it by the Frobenius norm which is $\sqrt{\sum_{i,j} |a_{ij}|^2}$.

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  • $\begingroup$ Thank you,@Robert Israel. It's useful, but this estimate is a bit coarse for computations $\endgroup$
    – cool
    Jan 5, 2015 at 23:23
  • $\begingroup$ Could you be a bit more precise about the type of estimate you are looking for? $\endgroup$ Jan 6, 2015 at 0:58
  • $\begingroup$ Sorry for my inaccurate question, i have edited the text $\endgroup$
    – cool
    Jan 6, 2015 at 1:06

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