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rec9 - EE221A Discussion 9 Lillian Ratli 1 Special Matrix...

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EE221A: Discussion 9 Lillian Ratliff October 29, 2010 1 Special Matrix Crash Course Definition U C n × n is unitary if U * U = I (orthogonal if U R n × n ) Definition H C n × n is hermitian if H = H * (self-adjoint), i.e. x, y h x, Hy i = h Hx, y i (symmetric if H R n × n ) FACT 1.0.1 Note: σ ( · ) denotes the spectrum. (a) σ ( U ) S 0 , i.e. λ σ ( U ) : | λ | = 1 PF: ( Uv = λv , v 6 = 0) ( v * U * = λ * v * ) ( v * U * Uv = λ * λv * v ) ( | v | 2 = | λ | 2 | v | 2 ) ( | λ | 2 = 1) | λ | = 1. (b) h Ux, Uy i = h x, y i ; | Ux | = | x | . (Its a total isometry of the whole space. It preserves length and angle.) PF: h Ux, Uy i = x * U * Uy = x * y = h x, y i . (c) Columns of U form an orthonormal basis for C n . PF: h Uu i , Uu j i = h e i , e j i = h u i , u j i = δ ij . or can just do U * U = I = ( u * 1 · · · u * n ) T ( u 1 · · · u n ) k u i k = 1. FACT 1.0.2 σ ( H ) R , Hv i = λ i v i , λ 1 6 = λ 2 ⇒ h v 1 , v 2 i = 0. PF: Hv = λv, v 6 = 0 ⇒ h v, Hv i = h Hv, v i = λ h v, v i = λ * h v, v i ⇒ λ = λ * λ R we just showed that all eigenvalues are real for a self-adjoint (hermitian) maps. h v 1 , Hv 2 i = λ 2 h v 1 , v 2 i = λ 1 h v 1 , v 2 i s FACT 1.0.3 U unitary such that U * HU = diag( σ ( H )), H = n 1 λ k u k u * k . PF: Using SVD, U, V, Σ : H = U Σ V * . Since H is self-adjoint, H = U Σ V * = V Σ U * (Σ is real-valued for
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