Linear Algebra and Optimization
Linear Algebra and Optimization Chapter One Problem 2.1. (1) To prove that ( H , ⋅ ) ({\cal H},\cdot) ( H , ⋅ ) is a group we must show that the following properties hold: Closure : if A , B ∈ H A,B \in {\cal H} A , B ∈ H then A ⋅ B ∈ H A \cdot B \in {\cal H} A ⋅ B ∈ H . To show this note that any upper triangular matrix A A A has the property [ 1 a b 0 1 c 0 0 1 ] = [ 1 a b d 1 c e f 1 ] ⊙ [ 1 1 1 0 1 1 0 0 1 ] \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & a & b \\ d & 1 & c \\ e & f & 1 \end{bmatrix} \odot \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} ⎣ ⎡ 1 0 0 a 1 0 b c 1 ⎦ ⎤ = ⎣ ⎡ 1 d e a 1 f b c 1 ⎦ ⎤ ⊙ ⎣ ⎡ 1 0 0 1 1 0 1 1 1 ⎦ ⎤ where ⊙ \odot ⊙ denotes the Hadamard product of two matrices and the first matrix is non-singular. In index notation we write this as A i j = a i j 1 j ≥ i ...