Wireless Ad Hoc and Sensor Networks

54 Wireless Ad Hoc and Sensor Networksvector norm by | ⋅|. When required to be specific, we denote the p-normnby | |. Recall that for any vector x ∈R ,⋅ pn∑ x ii=1| x| = | |1(2.8)⎛n⎞| x| p=| xi|p⎝⎜∑⎠⎟i=11p(2.9)| x| ∞= max| xi|i(2.10)The 2 -norm is the standard Euclidean norm.Given a matrix A, its induced p-norm is denoted by | A|.p LettingA= [ a ij ], recall that the induced 1-norm is the maximum absolute columnsummax| A| 1= ∑ | a ij |i(2.11)and the induced ∞-norm is the maximum absolute row sum| A| ∞= max ∑ | aij|ii(2.12)The induced matrix p-norm satisfies the inequality, for any vector x,| A| ≤ | A| | x|p p p(2.13)and for any two matrices A, B one also has| AB| ≤ | A| | B|p p p(2.14)Given a matrix A= [ a ij ], the Frobenius norm is defined as the root of thesum of the squares of all the elements:22 TA ≡ aij= tr A AF ∑ ( )(2.15)