## New answers tagged eigenvalues

3
votes

Accepted

### Significance of the length of the Perron eigenvector

That quantity $s = \frac{|u^Tv|}{\|u\|\|v\|}$ is the inverse of the eigenvalue condition number. The smaller it is, the more sensitive to perturbation the Perron value is.
More precisely, any ...

1
vote

### Sparse dense matrix versus non-sparse dense matrix in eigenvalue computation

If you take the transpose of the matrix, which has the same eigenvalues, and you divide by $w_0$, you get the (block) companion matrix of a quadratic
matrix polynomial with $n \times n$ coefficients. ...

2
votes

Accepted

### Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration

As Iosif Pinelis pointed out, you presumably meant
$$ \lambda_i^{-1} = \frac{e_i^T M e_i}{\lVert M e_i \rVert_2^2}, $$
where $e_i$ is the $i$th standard basis vector,
which is indeed a reasonable ...

2
votes

### Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration

$\newcommand{\La}{\Lambda}\newcommand{\la}{\lambda}$
No, your choice of $\La$ will not always lead to $S^k\to0$. In fact, for some symmetric positive-semidefinite matrices $M$, no choice of $\La$ ...

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