Search results
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
2 days ago · The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object.
v &\mapsto \norm{v} \end{split} \end{equation*} is a norm on \(V \) if the following three conditions are satisfied. Positive definiteness: \(\norm{v}=0 \) if and only if \(v=0\); Positive Homogeneity: \(\norm{av}=|a|\,\norm{v} \) for all \(a\in \mathbb{F} \) and \(v\in V\); Triangle inequality: \(\norm{v+w}\le \norm{v}+\norm{w} \) for all \(v ...
By a normed linear space (briefly normed space) is meant a real or complex vector space \ (E\) in which every vector \ (x\) is associated with a real number \ (|x|\), called its absolute value or norm, in such a manner that the properties \ (\left (\mathrm {a}^ {\prime}\right)-\left (\mathrm {c}^ {\prime}\right)\) of §9 hold.
The same condition number c = kAk kA−1k appears when the error is in the matrix. We have ∆A instead of ∆b in the error equation: Subtract Ax = b from (A + ∆A)(x + ∆x) = b to find A(∆x) = −(∆A)(x + ∆x). Multiply the last equation by A−1 and take norms to reach equation (7): k∆xk ≤ kA−1k k∆Ak kx + ∆xk. or.
In applied mathematics, Norms are functions which measure the magnitude or length of a vector. They are commonly used to determine similarities between observations by measuring the distance between them.
Jun 6, 2016 · Norm. A mapping $x\rightarrow\lVert x\rVert$ from a vector space $X$ over the field of real or complex numbers into the real numbers, subject to the conditions: $\lVert x\rVert\geq 0$, and $\lVert x\rVert=0$ for $x=0$ only; $\lVert\lambda x\rVert=\lvert\lambda\rvert\cdot\lVert x\rVert$ for every scalar $\lambda$; $\lVert x+y\rVert\leq\lVert ...
