# P r o j e c t s

N o t e s

# 1. Vector Norms

Let x=(x_1, cdots, x_n)^t be an ntimes 1 vector.

• p-norm: ||x||_p=left( sum_{i=1}^{n} |x_i|^p right)^{1/p}
• 1-norm: ||x||_1=sum_{i=1}^{n} |x_i|
• 2-norm(Euclidean norm): ||x||_2=left( sum_{i=1}^{n} |x_i|^2 right)^{1/2}
• infty-norm: ||x||_{infty}=max|x_i|
\begin{align} ||x||_{\infty}&= \lim_{p \to \infty} \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p} = \lim_{p \to \infty} \left( |x_j|^p \right)^{1/p} \   \text{where} \   j = \arg \max |x_i|  \\ &=\max|x_i|
\end{align}
• The graphs of ||x||_1= ||x||_2= ||x||_{infty}=1 for x in mathbb{R}^2 are as follows.

• For any vector bar{x}, ||bar{x} ||_{infty} leq || bar{x}||_2 leq || bar{x}||_1 . The right image shows this comparison when bar{x} in mathbb{R}^2.
• Meanwhile, ||x||_1 leq sqrt(n) ||x||_2, ||x||_2 leq sqrt(n) ||x||_{infty}, and ||x||_1 leq n ||x||_{infty}.

• If ||x||_p>0, then x ne 0.
• ||gamma x||_p=|gamma| ||x||_p where gamma in mathbb{R}.
• ||x+y||_p leq ||x||_p+||y||_p
• |||x||_p - ||y||_p  | leq ||x-y ||_p

# 2. Matrix Norms

Suppose that A is an m times n matrix and a_{ij} is the (i, j) element of A.
||A|| means the maximum stretching of A to any vector x.
\begin{align} \left\| A \right\| = \max_{x \ne 0} \frac{ \left\| Ax \right\| }{ \left\| x \right\| } \end{align}

• ||A||_1=max sum_{i=1}^m |a_{ij}|, which means the largest column sum of A.
• ||A||_2 is the largest singular value of A, which means the square root of the largest eigenvalue of A^tA.
• When A is symmetric, A^tAv=A(lambda v)=lambda^2 v where x and lambda are the eigenvector and eigenvalue of A. Then
\begin{align} \left| A \right|_2 = \sqrt{\lambda_{\max}(A^tA)} = \sqrt{\lambda_{\max}(A)^2} = |\lambda_{\max}(A)| \end{align}
• ||A||_{infty}=max sum_{j=1}^n |a_{ij}|, which means the largest row sum of A.
• If ||A||>0, then A ne O.
• ||gamma A||=|gamma| ||A|| where gamma in mathbb{R}.
• ||A+B|| leq ||A||+||B||
• ||AB|| leq ||A||||B||
• ||Ax|| leq ||A||||x||

︎Reference

[1] Michael T. Heath, Scientific Computing: An Introductory Survey. 2nd Edition, McGraw-Hill Higher Education.

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