Unformatted text preview: Notes on linear algebra (Monday 17¢h October, 2016, 23:10) page 68 Now, consider the following chain of equivalent statements33: (A is upper-triangular)
4:)» (AI-J- = 0 wheneveri > j)
(because this is how ”upper-triangular” is defined)
4:} (Aja = 0 whenever] >- i)
(here, we have just renamed i and j as j and i)
4:)» (Aja = 0 wheneveri < j) (because j > i is equivalent to f < j) 4:} ((ATij = 0 wheneveri < j) (here, we have replaced AH by (AT) . I, because of (53))
1,} 4:} (AT is lower-triangular) (because this is how ”lower-triangular” is defined) . Thus, Proposition 3.29 holds. D There are a few special classes of triangular matrices worth aivina names: ...
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