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So let us assume that C is lower-unitriangular. Our goal is to prove that C is a
product of lower addition matrices.
Let me intro
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0 Next, we similarly clear out all entries in the 3-rd column of the matrix, by
performing the downward row additions A113! 149,3:
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matrices Agkcfk with 1' > k; this corresponds to the a k downward row additions
that clear out the non-diagonal entries in the k
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From (53), we obtain
(Eu,e,n,m)f,j = 5i,u5j,v (59)
for all i 6 cfw_1,2,. . .,n andj E cfw_1,2, . . . ,m. (If this doesnt look lik
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Proof of Theorem 3.66. Theorem 3.62 (applied to C = A) shows that A is lower-
unitriangular if and only if A is a product of lowe
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Now, consider the following chain of equivalent statements35:
(A is upper-triangular)
4:) (AI'J = 0 wheneveri > j)
(because this
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from A13 2 0, we now derive At),- = 0 from (I.1 111)ij = 0). This is not
possible (after all, not every valid statement remains
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matrices are scaled entry by entry, we can therefore conclude how AEHIE, looks like:
Namely, AEHIU is the 11 >< n-matrix whose (
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Proof of Claim 2: Assume that In A is strictly upper-triangular. We must show
that A is upper-unitriangular.
We have assumed that
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Corollary 3.35. Let n E IN. Let A and B be two invertibly upper-triangular
n X n-rnatrices. Then, AB is also an invertibly upper-
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a Our proof of Proposition 3.31 (a) was really obvious; most of it was boilerplate
(writing down the assumptions, writing down th
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Example 3.30. (a) A 3 X 3-matrix is upper-unitriangular if and only if it has the
1 b 0
form 0 1 c for some b,c,c.
0 0 1
(b) A 3
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As I have said, the standard matrix units are building blocks for matrices: every
matrix can be obtained from them by scaling an
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Proposition 3.43. Let H E N, m E IN and p E N. Let u 6 cfw_1,2,.,r1 and
v E cfw_1,2,.,m. Let C be an m x p-matrix. Then, EHIC is
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Proof of Proposition 3.49. For every 1' 6 cfw_1,2,. . ., n and j 6 cfw_1,2, . . .,m, we have
u=lv=1
11 m
= Z Z (Au.vEu.v)i,j
u=lv
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[Hint In order to nd a right inverse of A, it is enough to find three column
vectors a, o, 30 (each of size 3) satisfying the equ
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for all i E cfw_1, 2,.,n and j E cfw_1,2,. . .,m (according to Proposition 3.47). Fur-
thermore, for anyi E cfw_1,2,. . .,n andj
Notes on linear algebra ( Thursday 29th September, 2016, 02:07) page 84
ilarly to how we found that C : Ail C in Step 1, we now obtain
C = A3110.
Step 3: Let us get rid of the (3, 2)-th entry of C by
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Example 3.46. Let us demonstrate Proposition 3.45 on a more spartanic example:
Let n = 2, m = 2 and p = 1. Let C = ( a b ) be a
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The proof of Proposition 3.57 is analogous.
Proof of Proposition 3.59. (a) The definition of A9,, yields Ag, 2 In | DEW = In |
v"
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Claim 1: If A is upper-unitriangular, then I,I A is strictly upper-triangular.
Claim 2: If I,I A is strictly upper-triangular, th
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Comparing this with
A3?! = H _|_ (Pt _I_ I) Eva? (by the definition of AH) ,
we obtain AinAllm = Anl . This proves Proposition 3
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(since matrices are scaled entry by entry).
We have
A33, (3 : (In + ism) C : IHC Heme = C + same.
V
:IH+AEM,IF :C
Hence, for each
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Theorem 3.66 is proven in both Cases 1 and 2;. this shows that Theorem 3.66 is
always valid. D
A similar result holds for upper-
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Note how Proposition 3.57 differs from Proposition 3.55: not only have rows
been replaced by columns, but also have u, and I) sw
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Theorem 3.24. Let n E ]N. Let A and B be two lower-triangular n X n-matrices.
(a) Then, AB is a lower-triangular n x n-matrix.
(b
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Case 2: We have 1' 75 a.
Case 3: We have neither i 7E a nor j 75 '0.
(In fact, it is possible that we are in Case 1 and Case 2 si
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are 1). Thus, every product of lower addition matrices is a product of lower-
unitriangular matrices, and thus itself must be lo
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3.8. The Aaddition matrices A3,
Now, we come to another important class of matrices (that can also be seen as
building blocks of