March 2014
Intermediate to advanced
412 pages
11h 16m
English
Content preview from Numerical Methods and Optimization
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Preliminaries 13
Definition 1.22 (Scalar Multiplication) Given a matrix A =[a
ij
]
m×n
and some scalar α ∈ IR , scalar multiplication αA is defined as
αA =
⎡
⎢
⎢
⎢
⎣
αa
11
αa
12
··· αa
1n
αa
21
αa
22
··· αa
2n
.
.
.
.
.
.
.
.
.
.
.
.
αa
m1
αa
m2
··· αa
mn
⎤
⎥
⎥
⎥
⎦
.
Next we list the algebraic rules of matrix addition and and scalar multi-
plication.
Theorem 1.4 Suppose that A, B, C are m ×n matrices, O is the m ×n
zero matrix, and α, β are scalars. The following rules of matrix addition
and scalar multiplication are valid:
(i) A + B = B + A commutativity
(ii) A + O = A additive identity
(iii) A +(−A)=O additive inverse
(iv) (A + B)+C = A +(B + C) associativity
(v) α(A + B)=αA + αB distributive pr
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