Courant-Fisher Theorem

The Courant – Fisher theorem is a theorem on the property of a Hermitian operator in a Hilbert function space . Also called minimax theorem ^[1] .

Content

Wording

\lambda _{k}=\sup \limits _{L_{k}}\inf \limits _{x\in L_{k}\cap S}{\frac {(Ax,x)}{(x,x)}}

{\ displaystyle \ lambda _ {k} = \ sup \ limits _ {L_ {k}} \ inf \ limits _ {x \ in L_ {k} \ cap S} {\ frac {(Ax, x)} {( x, x)}}}

A

{\ displaystyle A}

- a linear self-adjoint operator acting in a finite - dimensional complex or real space,

S

{\ displaystyle S}

- single sphere

e=e_{1}\dots e_{n}

{\ displaystyle e = e_ {1} \ dots e_ {n}}

- orthonormal basis of space

V

{\ displaystyle V}

consisting of eigenvectors of the operator

A

{\ displaystyle A}

,

\lambda _{k}

{\ displaystyle \ lambda _ {k}}

-

k

{\ displaystyle k}

operator eigenvalue

A

{\ displaystyle A}

and

\lambda _{1}\leq \lambda _{2}\leq \dots \leq \lambda _{n}

{\ displaystyle \ lambda _ {1} \ leq \ lambda _ {2} \ leq \ dots \ leq \ lambda _ {n}}

L_{k}

{\ displaystyle L_ {k}}

-

k

{\ displaystyle k}

-dimensional subspace

V

{\ displaystyle V}

.

Proof

$p=n-k+1$ ${\ displaystyle p = n-k + 1}$ ,
$L_{k}$ ${\ displaystyle L_ {k}}$ - $k$ ${\ displaystyle k}$ -dimensional subspace $V$ ${\ displaystyle V}$ ,
$W_{n-k+1}$ ${\ displaystyle W_ {n-k + 1}}$ - linear span of vectors $e_{k}\dots e_{n}$ ${\ displaystyle e_ {k} \ dots e_ {n}}$ .
$\dim L_{k}+\dim W_{n-k+1}=n+1$ ${\ displaystyle \ dim L_ {k} + \ dim W_ {n-k + 1} = n + 1}$ .
Whence it follows that $L_{k}\cap W_{n-k+1}\neq {\varnothing }$ ${\ displaystyle L_ {k} \ cap W_ {n-k + 1} \ neq {\ varnothing}}$ . Let be $x_{0}\in L_{k}\cap W_{n-k+1}$ ${\ displaystyle x_ {0} \ in L_ {k} \ cap W_ {n-k + 1}}$ and $\ \|x_{0}\|=1$ ${\ displaystyle \ \ | x_ {0} \ | = 1}$ .
Because $\lambda _{k}=\sup \limits _{x\in L_{k}\cap S}(Ax,x),$ ${\ displaystyle \ lambda _ {k} = \ sup \ limits _ {x \ in L_ {k} \ cap S} (Ax, x),}$ then ${\frac {(Ax_{0},x_{0})}{(x_{0},x_{0})}}\leq \lambda _{k}$ ${\ displaystyle {\ frac {(Ax_ {0}, x_ {0})} {(x_ {0}, x_ {0})}} \ leq \ lambda _ {k}}$ .
On the other hand: since $x_{0}\in L_{k}$ ${\ displaystyle x_ {0} \ in L_ {k}}$ then

\inf \limits _{x\in L_{k}\cap S}(Ax,x)\leq \lambda _{k}

{\ displaystyle \ inf \ limits _ {x \ in L_ {k} \ cap S} (Ax, x) \ leq \ lambda _ {k}}

\sup \limits _{L_{k}}\inf \limits _{x\in L_{k}\cap S}(Ax,x)\leq \lambda _{k}

{\ displaystyle \ sup \ limits _ {L_ {k}} \ inf \ limits _ {x \ in L_ {k} \ cap S} (Ax, x) \ leq \ lambda _ {k}}

Equality is achieved when $L_{k}=L({e_{1}\dots e_{k}})$ ${\ displaystyle L_ {k} = L ({e_ {1} \ dots e_ {k}})}$ .

Advanced

It's obvious that $\sup \limits _{L_{k}}\inf \limits _{x\in L_{k}\cap S}(Ax,x)=\inf \limits _{L_{n-k+1}}\sup \limits _{x\in L_{n-k+1}\cap S}(Ax,x)$ ${\ displaystyle \ sup \ limits _ {L_ {k}} \ inf \ limits _ {x \ in L_ {k} \ cap S} (Ax, x) = \ inf \ limits _ {L_ {n-k + 1 }} \ sup \ limits _ {x \ in L_ {n-k + 1} \ cap S} (Ax, x)}$ .

Notes

↑ Li Tsong-tao . Mathematical methods in physics. - M.: Mir, 1965. - c. 190

Literature

R. Bellman. Introduction to Matrix Theory
Lancaster. Matrix Theory
Prasolov Problems and theorems of linear algebra.
Ilyin, Kim. Linear Algebra and Analytical Geometry

[1] Li Tsong-tao . Mathematical methods in physics. - M.: Mir, 1965. - c. 190