- #1
Combinatorics
- 36
- 5
Homework Statement
Let [itex] \lambda_1 ,..., \lambda_n [/itex] be the eigenvalues of an [itex] nXn[/itex] self-adjoint matrix A, written in increasing order.
Show that for any [itex] m \leq n [/itex] one has:
[itex] \sum_{r=1}^{m} \lambda_r = min \{ tr(L) :dim(L) =m \} [/itex] where [itex] L [/itex] denotes any linear subspace of [itex] \mathbb {C} ^n [/itex], and [itex] tr(L):= \sum_{r=1}^{m} Q( \Phi_r) [/itex] for some orthonormal basis [itex] \{ \Phi _r \} [/itex] of [itex] L [/itex].
(Q is the quadratic form associated with the inner product).
Homework Equations
The Attempt at a Solution
I really have no idea on how to start this.
On the one hand, I think the trace will always be equal to m, which means I'm probably getting it wrong...
Hope you'll be able to help me
Thanks in advance