Chapter 1.6 SpecialMatrices

If a and b are both lower triangular then ab is lower

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Unformatted text preview: re both lower triangular, then AB is lower triangular. (c) A is invertible if and only if aii = 0 for all i. (d) If A is an invertible upper triangular matrix, then A−1 is upper triangular. If A is an invertible lower triangular matrix, then A−1 is lower triangular. Outline Diagonal Matrices Triangular Matrices Symmetric Matrices The Proof Proof. (a) AT = [bij ] where bij = aji . (b) If A and B are both upper triangular, then aij = 0 = bij whenever i > j . Let AB = [cij ]. Then if i > j , n aik bkj . cij = k =1 If 1 ≤ k < i , then aik = 0. If i ≤ k ≤ n, then k ≥ i > j so bkj = 0. Hence, aik bkj = 0 for 1 ≤ k ≤ n and so cij = 0. Hence, AB is upper triangular. (c) and (d) deferred for now. Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Example Example Discuss the Theorem using the following triangular matrices: 123 1 −2 4 A = 0 1 2 , B = 0 −1 3 000 0 02 Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Symmetric Matrices Deﬁnition Suppose A = [aij ] is an n × n m...
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