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Results tagged with perturbation-theory
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user 1898
2
votes
What is the relation between the eigenvectors of a sample covariance matrix and those of the...
It seems to me that the two bounds tell you two different, almost unrelated things.
The first focuses on what happens when the number of samples tends to infinity, and focuses on the distribution of …
- 20.2k
1
vote
Accepted
Eigenvalue Argument Perturbation
Uhm, it does not hold even in $\mathbb{R}^{1\times 1}$. $A=\varepsilon, B = -\varepsilon$ gives $LHS=\pi$, $RHS = 2\varepsilon$.
- 20.2k
3
votes
Accepted
Sensitivity of the range of a matrix
Quick self-answer for future reference: there is a brief statement of this result on Stewart, Sun, Matrix perturbation theory, p. 154:
If the hypotheses of Corollary 3.13 are satisfied (that is, w …
- 20.2k
3
votes
Can a perturbation of a matrix product always be represented as product of perturbations of ...
In numerical analysis lingo, you are more or less asking if matrix multiplication is backward stable. The answer seems to be no: see Section 3.5 of Higham, Accuracy and stability of numerical algorith …
- 20.2k
2
votes
Bounds on $\|(P+\Delta)^n - P^n\|_F$ for stochastic matrices
One possible starting point is via the telescopic sum
$$
\|(P+\Delta)^n-P^n\| = \|\sum_{i=0}^{n-1} (P+\Delta)^i ((P+\Delta) - P) P^{n-1-i}\| \leq \|(P+\Delta)^i\| \|\Delta\| \|P^{n-1-i}\|.
$$
Since al …
- 20.2k