NUMERICAL-ANALYSIS Course

Norms

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A norm is a measure of an error. We got 3 different notations for norms according to what is inside the vertical bars.

  • \(\mid \cdot \mid \) : norm of a real/complex
  • \(\mid\mid \cdot \mid\mid\) : norm of a vector
  • \(\mid\mid\mid \cdot \mid\mid\mid\) : norm of a matrix

The rules/properties of a norm are

  • \( N(x) \ge 0 \)
  • \( N(x+y) \le N(x)+N(y) \)
  • \( N(x) = 0 \Leftrightarrow x = 0 \)
  • \( N(\lambda{x}) = |\lambda| N(x) \)

Norms in 1, 2, infinity

You will see a lot of norms with a small index. These are their formulas

\[ ||x||_1 = \sum_{i=1}^{n}{ |\ x_i |} \]

\[ ||x||_2 = (\sum_{i=1}^{n}{ |\ x_i |^2} )^{1/2} = \sqrt{\sum_{i=1}^{n}{ |\ x_i |^2}} \]

\[ ||x||_{+\infty } = \max_{i \in \mathbb{[}1:n\mathbb{]}} | \ x_i |^2 \]


Some changes possibles

Just in case you want to change your expression to another one, then here is some help

  • \( ||v||^2_2 = v^t * v \)
  • \( \rho(A) \le ||A|| \) (rho is the highest eigenvalue)
  • \( ||Ax|| \le |||A||| * ||x|| \)
  • \( ||AQ||_2 = ||QA||_2 = ||A||_2 \)
  • \( ||Qx||^2_2 =||x||^2_2 \)

And you must never forget this one

\[ |||A|||_2 = \max_{ y \neq 0 } \frac{||Ay||_2}{||y||_2} \]


Examples

Q: Demonstrate \(||Qx||^2_2 =||x||^2_2\)

\begin{split} ||Qx||^2_2 \Leftrightarrow (Qx)^* * Qx \Leftrightarrow \\ x^t * Q^t * Q * x \Leftrightarrow x^t * x \Leftrightarrow \\ ||x||^2_2 \end{split}

Note that Q^t is the matrix transpose of a matrix $Q \in \mathbb{R}^{n}$. We would replace this by $Q^*$ with $Q \in \mathbb{C}^{n}$. Also \( Q^t * Q = Id \) so I removed $Q$ by using this property.

Q: Demonstrate \(||AQ||_2 = |||A|||\)

\begin{split}||AQ||_2 := \max_{ x \neq 0 } \frac{||AQx||_2}{||x||_2} \\ with \ y = Qx \\ \Leftrightarrow \max_{ x \neq 0 } \frac{||Ay||_2}{||y||_2} := |||A|||\end{split}