 Courant–Fischer min-max principle

ID: courant-fischer-min-max-principle

For a compact positive self-adjoint operator

T

with decreasing eigenvalues

λ_{1} \geq λ_{2} \geq \dots

, the Courant–Fischer principle gives

λ_{k} = min_{d i m V = k - 1} max_{x ⊥ V ∥ x ∥ = 1} ⟨ T x, x ⟩ .

In particular,

S \geq T

in quadratic-form order implies

λ_{k} (S) \geq λ_{k} (T)

for every

k

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