3M1 Mathematical Methods Tripos Revision

$\require{amstext} \require{amsmath} \require{amssymb} \require{amsfonts} \newcommand{\norm}[1]{\lVert #1 \rVert}$

Linear Algebra

Conjugate Transpose: $M^H = \overline{M^T} = \overline{M}^T$
Hermitian: if M^H = M
- $(AB)^H = B^H A^H$
- $A^H A$ must be hermitian
Unitary matrix: if $M^H = M^{-1}$
Hermitian positive definite Matrix: $x^H M x > 0 \forall x \in \mathcal{C}^n\backslash 0$

Vector $l_p$ norms: $\norm{x} = (\sum_{i} |x_i|^p )^{1/p}$
- Infinite norm just find the maximum term of the vector $\norm{x}_{\infty} = \max_i |x_i|$ which is also knownas the maiximum norm.
Matrix induced norm: $\norm{x}^2_{A} = x^H A x$
Properties:
- Linearity: $\norm{kx} = |k| \norm{x}$
- Triangle inequality: $\norm{x+y} \le \norm{x} + \norm{y}$
- Another inequality: $\norm{xy} \le \norm{x}\norm{y}$

Operator norms $\norm{A} = \max_{\forall x \in \mathcal{C}^n\backslash 0} \frac{\norm{Ax}}{\norm{x}}$
- This norm measures the maximum amount by which the matrix $A$ can re-scale a vector $x$
1-norm: $\norm{A} =\max_j \sum_{i=1}^n |a_{ij}|$ which is column of A with maximum $l_1$ norm
$\infty$ norm: $\norm{A} = \max_i \sum_{j}^n|a_{ij}|$ which is row of A with maximum $l_1$ norm
$l_2$ norm: $\norm{A} = \sqrt{\lambda_{\max}(A^H A)}$

$\kappa(A) = \norm{A}\norm{A^{-1}}$
For the 2-norm: \kappa_2(A) = \frac{\sqrt{\lambda_{\max}A^H A}}{\sqrt{\lambda_{\min}A^H A}}
- which is the max singular value over min singular value
Since eigenvalue is reciprocal for matrix inverse, if A is Hermitian, then: $\kappa_2(A) = \frac{|\lambda(A)|max}{|\lambda(A)|min}$
A matrix with a large condition number is ill-conditioned, which lead to instable computation $\rightarrow$ small error lead to large computation error