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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 64 I for some numbers a, la", a”, am. Here is something obvious: Proposition 3.22. Let n E N. (a) An a X a-matrix A is diagonal if and only if A is both upper-triangular and lower-triangular. (b) The zero matrix on x” and the identity matrix In are upper-triangular, lower- triangular and diagonal. A less trivial fact is that the product of two upper-triangular matrices is upper- triangular again. We shall show this, and a little bit more, in the following theorem: Theorem 3.23. Let a E N. Let A and B be two upper-triangular a x a-matrices. (a) Then, AB is an upper-triangular a x n-matrix. (b) The diagonal entries of AB are (AB)i,i = ALI-Biri fOI' all If 6 {1,2, . . . , fl} . (c) Also, A + B is an upper-triangular a x a-matrix. Furthermore, AA is an upper-triangular matrix whenever it is a number. Note that Theorem 3.23 (b) savs that each diagonal entrv of AB is the oroduet of ...
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