rec9 - EE221A: Discussion 9 Lillian Ratliff October 29,...

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Unformatted text preview: EE221A: Discussion 9 Lillian Ratliff October 29, 2010 1 Special Matrix Crash Course Definition U ∈ Cn×n is unitary if U ∗ U = I (orthogonal if U ∈ Rn×n ) Definition H ∈ Cn×n is hermitian if H = H ∗ (self-adjoint), i.e. ∀x, y H ∈ Rn×n ) FACT 1.0.1 Note: σ (·) denotes the spectrum. (a) σ (U ) ⊂ S , i.e. ∀λ ∈ σ (U ) : |λ| = 1 PF: (U v = λv , v = 0) ⇒ (v ∗ U ∗ = λ∗ v ∗ ) ⇒ (v ∗ U ∗ U v = λ∗ λv ∗ v ) ⇒ (|v |2 = |λ|2 |v |2 ) ⇒ (|λ|2 = 1) ⇒ |λ| = 1. (b) U x, U y = x, y ; |U x| = |x|. (Its a total isometry of the whole space. It preserves length and angle.) PF: U x, U y = x∗ U ∗ U y = x∗ y = x, y . x, Hy = H x, y (symmetric if (c) Columns of U form an orthonormal basis for Cn . PF: U ui , U uj = ei , ej = ui , uj = δij . or can just do U ∗ U = I = (u∗ · · · u∗ )T (u1 · · · un ) ⇒ n 1 ui = 1. FACT 1.0.2 σ (H ) ⊂ R, Hvi = λi vi , λ1 = λ2 ⇒ v1 , v2 = 0. PF: Hv = λv, v = 0 ⇒ v , Hv = H v, v = λ v , v = λ∗ v , v ⇒ λ = λ∗ ⇒ λ ∈ R we just showed that all eigenvalues are real for a self-adjoint (hermitian) maps. v1 , Hv2 = λ2 v1 , v2 = λ1 v1 , v2 s FACT 1.0.3 ∃U unitary such that U ∗ HU = diag(σ (H )), H = n λk uk u∗ . 1 k PF: Using SVD, ∃U, V, Σ : H = U ΣV ∗ . Since H is self-adjoint, H = U ΣV ∗ = V ΣU ∗ (Σ is real-valued for Å ã Σ0 ∗ SVD). Thus, we can take U = V . So recall that H = (U1 U2 ) (V1∗ V2∗ )T = U1 Σ V1∗ = V1 Σ U1 . 00 So, v = U ej : Hv = U ΣU ∗ U ej = U Σej = γj U ej = γj v . FACT 1.0.4 sup v =1 v ∗ Hv = max σ (H ), inf v =1 v ∗ Hv = min σ (H ). PF: w = U ∗ v s.t. v ∗ Hv = v ∗ U ΣU ∗ v = w∗ Σw ⇒ e∗ Σej = γj . j Definition P = P ∗ ≥ 0 is positive (resp semi-) defintite if ∀v : v ∗ P v ≥ 0 ; P > 0 if ∀v = 0 : v ∗ P v > 0. 1 Definition P = P ∗ P ≥ 0 ⇔ ∀λ ∈ σ (P ) : λ ≥ 0 : P > 0 ⇔ ∀λ ∈ σ (P ): λ > 0. FACT 1.0.5 P = P ∗ > 0: x, y := x∗ P y = x, P y is an innerproduct, √ x∗ P x is a norm. P Definition Jordan Form: 2 ...
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This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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rec9 - EE221A: Discussion 9 Lillian Ratliff October 29,...

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