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Unformatted text preview: 1 LECTURE 3.1A Dr. Vyas Overview: (Covers section 3.3) • Upper Triangular Linear Systems • Back Substitution • Examples/Demos UPPER TRIANGULAR LINEAR SYSTEMS 1. Definition: An N N × matrix [ ] i j N N a × = A is called upper triangular provided that the elements satisfy 0, i j a for all i j = . The N N × matrix [ ] i j N N a × = A is called lower triangular provided that the elements satisfy 0, i j a for all i j = < . 2. If A is an upper triangular matrix, then = A X B is said to be an upper triangular system of linear equations and has the form, 11 12 13 1 1 1 1 1 22 23 2 1 2 2 2 3 3 33 3 1 3 1 1 1 1 1 N N N N N N N N N N N N N N NN a a a a a x b a a a a x b x b a a a x b a a x b a-------- = KK KK KK M M M M MKKKMKKKM KK KKK 3. The lower triangular system will have non-zero terms below the diagonal of the matrix while zeroes above the diagonal, unlike the case above. 4. Theorem 3.5 : Suppose that = A X B is an upper triangular system with the form given above. If kk a ≠ for 1, 2, ....., k N = , then there exists a unique solution to the upper triangular system. The unique solution is found by first determining / N N NN x b a = and then using a backward substitution algorithm given as, 1 1, 2, ........,1 N k k j j j k k kk b a x x for k N N a...
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