On Two Conjectures Regarding Eigenvalue Perturbations And a Common Counterexampl
On Two Conjectures Regarding Eigenvalue Perturbations And a Common Counterexampl
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The proof is a simple computation.
-1 -B -B 2 -1 -B -1 I ■ i) ■ ; . .
L0 = II* 1 ! I - 1 _ -1 /. ' -1 B - 1 1+X 1-1 » 1 -fl 2 (1+ ^ 2 ) 2-v3/2 -1 -fl -1 ■ -B" -1 -B -1 for fi»l. When X = i>#0, a mta (A-/|fc/) = ||(A-/m. /)-'||-' : (l + «p. ) 3 |n|fl 2 which as a function of n, reaches its minimum 3 3/2 / (2 fl 2 ) at ji = 2 _1/2 . If we let A be n by n and of the same structure as before: A = then o- min (A) ~ B ' as before and o- min (A-i>/) achieves its minimum 0(S' ") for \l ...= 0(\). Thus for large B and/or large n, using Conjectures 1 and 2 as computational heuristics can lead to very bad results.
Note however that the matrix A is quite special: not only is it defective but it is nearly derogatory. Of course perturbing A slightly would yield a matrix with distinct eigenvalues with similarly shaped S(A, (. ), so defectiveness per se is not essential, but nearness to a defective and derogatory matrix.
What to read after On Two Conjectures Regarding Eigenvalue Perturbations And a Common Counterexampl?
You can find similar books in the "Read Also" column, or choose other free books by James Weldon Demmel to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The proof is a simple computation.
-1 -B -B 2 -1 -B -1 I ■ i) ■ ; . .
L
Note however that the matrix A is quite special: not only is it defective but it is nearly derogatory. Of course perturbing A slightly would yield a matrix with distinct eigenvalues with similarly shaped S(A, (. ), so defectiveness per se is not essential, but nearness to a defective and derogatory matrix.
What to read after On Two Conjectures Regarding Eigenvalue Perturbations And a Common Counterexampl?
You can find similar books in the "Read Also" column, or choose other free books by James Weldon Demmel to read onlineMoreLess
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