Lecture 28 Monday March 30 Matrix Norms and Eigen values

Lecture 28 Monday March 30 Matrix Norms and Eigen values -...

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Unformatted text preview: Lecture 28: Matrix Norms and Eigen values Matrix Norms Frobenius norm a i, j 2 ij 2 2-norm j u j max|v|1 | Av | 1 2 12 2 32 ... |A|2 = |UΣVT|2 = |Σ|2 = 1 |A|F = |UΣVT|F = |Σ|F = Theorem: (1)n ( n (a11 a22 ... ann ) n 1 ...) |A – Ak|F = (1) n (a11 a22 ... ann ) (1)n (11 22 ... nn ) min rank ( B )k | A B |F c(k ) 2 |A – A2|2 = min rank ( B ) k | A B |2 Let B be a square matrix. Find solutions of the form: B x = x Consider B = AAT = ( = i T jii j T j k 1 2 d u v )( u v i 1 T iii n n u v v u = i, j j 1 j j T j ) 2 T i ii uu Buj = i i 2 T i ii u u u j = 2u j j * ui uiT 1 when i j B is symmetric and positive semi definite xTBx 0 xTAATx = (ATx)2 0 What does the probability distribution of Eigen values of random matrix look like? (Wigner, 1957) The distribution of eigenvectors can tell you whether a matrix is random. Lemma: trace(A) = λ1 + λ2 + λ3 + … + λn trace(A2) = λ12 + λ22 + λ32 + … + λn2 Proof: There exists a nontrivial solution for x in (A – λI)x = 0 only if det(A - λI) = 0 det(A - λI) is polynomial in λ. (1) n ( ) (1) ( n i 1 i n n (1 2 ... n ) n1 ...) = (a11 – λ)det(n – 1) + … = (a11 – λ) (a22 – λ) (a33 – λ)… (ann – λ) + … = (1)n ( n (a11 a22 ... ann ) n1 ...) We want to equate λn-1 coefficients. The coefficient for λn-1 is (1)n (a11 a22 ... ann ) (1)n (a11 a22 ... ann ) (1)n (11 22 ... nn ) We want to show P(λ) is like 1 2 (semicircle) Normalize the Eigen values to range [-1, 1] 2 1 2 Find equal moments for both equations to show equality Kth moment: c( k ) 2 k 1 2 d ...
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Lecture 28 Monday March 30 Matrix Norms and Eigen values -...

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