MatrixNorms - Silvana Ilie MTH510 Lecture Notes 1 Matrix...

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Silvana Ilie - MTH510 Lecture Notes 1 Matrix Norms and Condition A vector X = [ x 1 , x 2 , x 3 , . . . , x n ] has the Euclidian norm k X k 2 = v u u t n X i =1 x 2 i = q x 2 1 + x 2 2 + . . . + x 2 n For an n × n –matrix A k A k f = v u u t n X i =1 n X j =1 a 2 ij is called the Frobenius norm . 1.1 Additional Norms for Vectors k X k p = ˆ n X i =1 | x i | p ! 1 /p for case p = 2 ⇒ k X k 2 is the Euclidean norm for case p = 1 ⇒ k X k 1 = n X i =1 | x i | for case p = 2 ⇒ k X k = max i =1 ,...n | x i | the largest absolute value of an element in X 1.2 Additional Norms for Matrices Norm-1 or the column-sum norm : k A k 1 = max j =1 ,...n n X i =1 | a ij | Norm- or the row-sum norm : k A k = max i =1 ,...n n X j =1 | a ij | Example Compute k A k 1 and k A k where A = 3 . 3330 15920 - 10 . 333 2 . 2220 16 . 710 9 . 6120 1 . 5611 5 . 1791 1 . 6852 1
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Solution k A k 1 = max { 3 . 3330 + 2 . 2220 + 1 . 5611 , 15920 + 16 . 710 + 5 . 1791 , | - 10 . 333 | + 9 . 6120 + 1 . 6852 } = 15941 . 8891 k A k = max { 3 . 3330 + 15920 + | - 10 . 333 | , 2 . 2220 + 16 . 710 + 9 . 6120 , 1 . 5611 + 5 . 1791 + 1 . 6852 } = 15933 . 660 1.3 Norm-2 of a Matrix Let μ max be the largest eigenvalue of A T A , we define k A k 2 = ( μ max ) 1 / 2 to be the spectral norm of A .
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