A Conjecture about eigenvalues
Peter B
Let A be a diagonal p-dim matrix with a_1 >= a_2 >=...>= a_p >=0 on the
diagonal. For an arbitrary column vector v, the matrix (A + v*t(v))
has the eigenvalues b_1>=b_2>=...>=b_p>=0 such that b_i >= a_i for all
i=1,..,p.
Note: t(v) is a transpose of v, that is, a row vector (so, v*t(v) is a
matrix).
I can prove it only in some special cases, but not in general. Please
let me know, if you have seen anything similar, or know how to prove
it.
Ilya Zakharevich
Peter B
<pio...@gmail.com>], who wrote in article <gb8fe2$m75$1...@news.acm.uiuc.edu>:
Look for minimax characterization of eigenvalues of self-adjoint
operators.
Hope this helps,
Ilya
Peter B
> [A complimentary Cc of this posting was sent to
> Peter B
>
> > I'm looking for a proof of the following Conjecture
>
> > Let A be a diagonal p-dim matrix with a_1 >= a_2 >=...>= a_p >=0 on the
> > diagonal. For an arbitrary column vector v, the matrix (A + v*t(v))
> > has the eigenvalues b_1>=b_2>=...>=b_p>=0 such that b_i >= a_i for all
> > i=1,..,p.
>
> > Note: t(v) is a transpose of v, that is, a row vector (so, v*t(v) is a
> > matrix).
>
> > I can prove it only in some special cases, but not in general. Please
> > let me know, if you have seen anything similar, or know how to prove
> > it.
>
> Look for minimax characterization of eigenvalues of self-adjoint
> operators.
>
> Hope this helps,
> Ilya
Thanks.
The question was answered in sci.math. It can be concluded from the
interlocking eigenvalues lemma used in optimization theory. For
example see p. 300 in David G. Luenberger and Yinyu Y, Linear and
Nonlinear Programming: Third Edition (2008). You can also find the
proof of my version at