By Jean Gallier
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Additional info for Applications of Scientific Computation [Lecture notes]
Here, i = √ −1. Since x · x ≥ 0 (property (a)), the square root of x · x makes sense, and we define the (Euclidean) length or (Euclidean) norm of x as √ x = x · x = (x21 + x22 + · · · + x2n )1/2 . Remark: The Euclidean norm is also denoted by x 2 , and is sometimes called the 2-norm. In view of property (b), x = 0 iff x = 0. Another useful property of the Euclidean norm is this: For every scalar λ ∈ R, for every vector x ∈ Rn , λx = |λ| x . The absolute value is needed because λ could be negative, but a norm is always nonnegative.
The last two properties say that the inner product is positive definite. Using the first five properties, we can show the following useful fact: λx + µy 2 = (λx + µy) · (λx + µy) = λ2 x 2 + 2λµ x · y + µ2 y 2 . In particular, for λ = µ = 1 we get x+y 2 = x 2 + 2x · y + y 2 . This shows that Pythagoras Law holds, namely x+y 2 = x 2 + y 2 iff x · y = 0, that is iff x and y are orthogonal. We are now ready to prove an important property of inner products, the Cauchy-Schwarz inequality. This property implies that the Euclidean norm satisfies what is known as the triangle inequality, another crucial property.
Ak = . akk k · · · akk n .. . akn k · · · akn n Actually, note akij = aii j for all i, j with 1 ≤ i ≤ k − 2 and i ≤ j ≤ n, since the first k − 1 rows remain unchanged after the (k − 1)th step. We will prove later that det(Ak ) = ± det(A). Consequently, Ak is invertible. The fact that Ak is invertible iff A is invertible can also be shown without determinants from the fact that there is some invertible matrix Mk such that Ak = Mk A, as we will see shortly. Since Ak is invertible, some entry akik with k ≤ i ≤ n is nonzero.