Vector/Matrix Norms and Iterative Linear Solvers

a. ‖x‖ ≥ 0 for all x ∈ IRn, and ‖x‖ = 0 if and only if x is the zero vector. b. ‖αx‖ = |α|‖x‖. c. ‖x+ y‖ ≤ ‖x‖+ ‖y‖ [called the “triangle inequality”]. While this may seem a bit esoteric, consider that if x ∈ IR1, i.e. if x is a real number, the absolute value function, | · |, satisfies all of the above properties, and therefore constitutes as a norm. Another familiar norm would be the Euclidean norm for vectors x ∈ IR2. Example 1 (Euclidean norm on IR2). Consider the vector [x, y] ∈ IR2. Then the function ‖ [x, y] ‖ = √ x2 + y2