Chapter 1.6 SpecialMatrices

Demonstrate the previous theorem by computing ac and

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Unformatted text preview: evious Theorem by computing AC and DA, where −1 5 7 2 5 −3 7 9 10 −8 8 −5 , D = C = 0 1 −3 0 −4 30 6 0 6 −7 9 Outline Diagonal Matrices Triangular Matrices Diagonal Matrices Definition Suppose A = [aij ] is an n × n matrix. (a) A is upper triangular if aij = 0 whenever i > j . (b) A is lower triangular if aij = 0 whenever i < j . (c) A is triangular if A is either upper or lower triangular. a11 a12 a13 a21 a22 a23 a31 a32 a33 Diagonal matrices are both upper and lower triangular Symmetric Matrices Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Properties of Triangular Matrices Theorem Suppose A = [aij ] and B = [bij ] are n × n triangular matrices. (a) If A is upper triangular, then AT is lower triangular. If A is lower triangular, then AT is upper triangular. (b) If A and B are both upper triangular, then AB is upper triangular. If A and B a...
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